{"id":8338,"date":"2026-03-19T14:43:44","date_gmt":"2026-03-19T14:43:44","guid":{"rendered":"https:\/\/mrenglishkj.com\/?p=8338"},"modified":"2026-03-26T03:03:13","modified_gmt":"2026-03-26T03:03:13","slug":"sat-2026-math-test-get-free-access-of-module-2","status":"publish","type":"post","link":"https:\/\/us.mrenglishkj.com\/sat\/sat-2026-math-test-get-free-access-of-module-2\/","title":{"rendered":"SAT 2026 Math Test (Get Free Access of Module 2"},"content":{"rendered":"\n<h2 class=\"wp-block-heading\">Practice the 2026 SAT Math Test &#8211; Module 2nd All Solutions with Desmos Key Points<\/h2>\n\n\n\n<p>How was your Module 1st? How much have you scored? Please tell us in the comment. The SAT math seems tough without Desmos Calculator. Fear not, we have provided you solutions of all questions with Desmos tricks. This test is a practice test of 2026 SAT Math Module Second. The best parts are:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>solutions of all questions,<\/li>\n\n\n\n<li>step-by-step explanations,<\/li>\n\n\n\n<li>how to verify the correct answer,<\/li>\n\n\n\n<li>description of correct and incorrect options,<\/li>\n\n\n\n<li>tips and tricks,<\/li>\n\n\n\n<li>and Desmos Calculator Hacks.<\/li>\n<\/ul>\n\n\n\n<p>Like the other exams, it has the same format and all the necessary features for you to become a SAT master in math. You just take the Module 2nd exam to practice your skills. The best part is that you practice within the time limit, and there are explanations of answers, tips and tricks to get a perfect score on the SAT. You will find Math easy after this.<\/p>\n\n\n\n<div style=\"height:70px\" aria-hidden=\"true\" class=\"wp-block-spacer\"><\/div>\n\n\n\n<figure class=\"wp-block-image size-full is-style-rounded has-lightbox\"><img loading=\"lazy\" decoding=\"async\" width=\"1024\" height=\"1024\" src=\"https:\/\/us.mrenglishkj.com\/sat\/wp-content\/uploads\/sites\/2\/2025\/12\/4538a93b-8d99-4192-b6d2-d48fe56b959f.png\" alt=\"Take the SAT Math test of 2025 with all four options solutions and math tricks with desmos calculator hack\" class=\"wp-image-8290\" srcset=\"https:\/\/us.mrenglishkj.com\/sat\/wp-content\/uploads\/sites\/2\/2025\/12\/4538a93b-8d99-4192-b6d2-d48fe56b959f.png 1024w, https:\/\/us.mrenglishkj.com\/sat\/wp-content\/uploads\/sites\/2\/2025\/12\/4538a93b-8d99-4192-b6d2-d48fe56b959f-300x300.png 300w, https:\/\/us.mrenglishkj.com\/sat\/wp-content\/uploads\/sites\/2\/2025\/12\/4538a93b-8d99-4192-b6d2-d48fe56b959f-150x150.png 150w, https:\/\/us.mrenglishkj.com\/sat\/wp-content\/uploads\/sites\/2\/2025\/12\/4538a93b-8d99-4192-b6d2-d48fe56b959f-768x768.png 768w\" sizes=\"auto, (max-width: 1024px) 100vw, 1024px\" \/><\/figure>\n\n\n\n<div style=\"height:70px\" aria-hidden=\"true\" class=\"wp-block-spacer\"><\/div>\n\n\n\n<h3 class=\"wp-block-heading\">ABOUT THE SAT MODULES<\/h3>\n\n\n\n<p>The SAT is divided into four modules. There are two categories with each split into two modules. The first category is &#8220;Reading and Writing&#8221; with two modules. The second category is &#8220;Math&#8221; with two modules. The one, you will do below is SAT Math 2026 Practice Test Module 2nd.<\/p>\n\n\n\n<p>The first module has questions ranging from easy to difficult, but the second module only contains medium to difficult questions, no easy. If you want to take some other SATs, visit the links below.<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li><a href=\"https:\/\/us.mrenglishkj.com\/sat\/category\/sat-english\/module-1st\/\" target=\"_blank\" rel=\"noopener\" title=\"\">1st Module of SAT Reading And Writing Practice Tests<\/a><\/li>\n\n\n\n<li><a href=\"https:\/\/us.mrenglishkj.com\/sat\/category\/sat-english\/module-2nd\/\" target=\"_blank\" rel=\"noopener\" title=\"\">2nd Module of SAT Reading And Writing Practice Tests<\/a><\/li>\n\n\n\n<li><a href=\"https:\/\/us.mrenglishkj.com\/sat\/category\/sat-math\/1st-module\/\" target=\"_blank\" rel=\"noopener\" title=\"\">1st Module of SAT Math Practice Tests<\/a><\/li>\n\n\n\n<li><a href=\"https:\/\/us.mrenglishkj.com\/sat\/category\/sat-math\/2nd-module\/\" target=\"_blank\" rel=\"noopener\" title=\"\">2nd Module of SAT Math Practice Tests<\/a><\/li>\n<\/ul>\n\n\n\n<div style=\"height:70px\" aria-hidden=\"true\" class=\"wp-block-spacer\"><\/div>\n\n\n\n<h3 class=\"wp-block-heading\">THE SAT MATH MODULE 2ND<\/h3>\n\n\n\n<p>The second module of Math in SAT contains four segments: &#8220;Algebra,&#8217; &#8216;Advanced Math,&#8217; &#8216;Problem-Solving and Data Analysis,&#8217; and &#8216;Geometry and Trigonometry.&#8221; The questions in Module 2nd are from medium to difficult. In a real SAT exam, you must answer 22 questions within 35 minutes. We have provided you with the same in this Practice Test.<\/p>\n\n\n\n<div style=\"height:70px\" aria-hidden=\"true\" class=\"wp-block-spacer\"><\/div>\n\n\n\n<h4 class=\"wp-block-heading\">Instructions for the SAT Real-Time Exam: Tips Before Taking Tests<\/h4>\n\n\n\n<ol class=\"wp-block-list\">\n<li><strong>Previous-and-Next:<\/strong> Like in real SAT exam, you can move freely from one question to another, same things you can do here. You select one option and move forward but you realized something, so you came back and change your option. You can do that here and in the real SAT exam too.<\/li>\n\n\n\n<li><strong>Timer: <\/strong>On the top of the slide, you will see the timer, it starts from 0 and for Module 1st of Math you will get <strong><em>35 minutes to finish 22 questions<\/em><\/strong>. Always try to finish the test before 35 minutes.<\/li>\n\n\n\n<li><strong>Image:<\/strong> You can click on a graph, table, or other image to expand it and view it in full screen.<\/li>\n\n\n\n<li><strong>Mobile:<\/strong> You cannot take the real exam on mobile, but our practice exam you can take on mobile phone.<\/li>\n\n\n\n<li><strong>Calculator<\/strong>: Below the Test, you will see a Desmos calculator and graph for Math. The same, Desmos, will be used in real exams, so learn &#8220;How to use Desmos Calculator.&#8221;<\/li>\n\n\n\n<li><strong>Answer All<\/strong>: Even if you do not know the correct answer of a question, still guess it because there is no Negative marking.<\/li>\n\n\n\n<li><strong>Last Questions<\/strong>: The harder the question, the more marks it will fetch for you. So most likely, you will find later question difficult and more time-consuming, so utilize your time accordingly.<\/li>\n\n\n\n<li><strong>Tips:<\/strong> This article will help you learn more about the SAT Exams. <a href=\"https:\/\/us.mrenglishkj.com\/sat\/everything-about-the-sat\/\" target=\"_blank\" rel=\"noopener\" title=\"SAT: EVERYTHING ABOUT THE SAT\">SAT: EVERYTHING ABOUT THE SAT<\/a><\/li>\n<\/ol>\n\n\n\n<div style=\"height:70px\" aria-hidden=\"true\" class=\"wp-block-spacer\"><\/div>\n\n\n\n        <script>\n          window.KQ_FRONT = window.KQ_FRONT || {};\n          window.KQ_FRONT.quiz_id = 5;\n          window.KQ_FRONT.rest = \"https:\/\/us.mrenglishkj.com\/sat\/wp-json\/kq\/v1\/\";\n        <\/script>\n        <div id=\"kapil-quiz-5\"\n             class=\"kapil-quiz-container\"\n             data-kq-app\n             data-quiz-id=\"5\">\n            <div class=\"kq-loading\">Loading quiz...<\/div>\n        <\/div>\n        \n    <div id=\"kq-auth-modal\" class=\"kq-auth-modal\" style=\"display:none;\">\n      <div class=\"kq-auth-modal-inner\">\n        <button id=\"kq-auth-close\" class=\"kq-auth-close\" aria-label=\"Close\">\u2716<\/button>\n\n        <!-- TAB NAV -->\n        <div class=\"kq-auth-tabs\" role=\"tablist\">\n          <button class=\"kq-tab active\" data-tab=\"register\" type=\"button\" role=\"tab\" aria-selected=\"true\">Register<\/button>\n          <button class=\"kq-tab\" data-tab=\"login\" type=\"button\" role=\"tab\" aria-selected=\"false\">Login<\/button>\n          <button class=\"kq-tab\" data-tab=\"forgot\" type=\"button\" role=\"tab\" aria-selected=\"false\">Forgot<\/button>\n        <\/div>\n\n        <!-- PANELS -->\n        <div class=\"kq-auth-panel-wrap\">\n\n          <!-- REGISTER -->\n          <div class=\"kq-auth-panel\" data-panel=\"register\" style=\"display:block\">\n            <div class=\"kq-auth-card\">\n              <h3>Register<\/h3>\n              <div class=\"kq-field\">\n                <input id=\"kq-signup-username\" placeholder=\"Username\" \/>\n              <\/div>\n              <div class=\"kq-field\">\n                <input id=\"kq-signup-email\" placeholder=\"Email\" type=\"email\" \/>\n              <\/div>\n              <div class=\"kq-field\">\n                <input id=\"kq-signup-password\" placeholder=\"Password\" type=\"password\" \/>\n                <button class=\"kq-toggle-pass\" type=\"button\" aria-label=\"Toggle password\">\ud83d\udc41<\/button>\n              <\/div>\n              <button id=\"kq-signup-btn\" class=\"button kq-btn-small\">Register<\/button>\n              <small style=\"display:block;margin-top:8px;\">Already registered? Use Login tab.<\/small>\n            <\/div>\n          <\/div>\n\n          <!-- LOGIN -->\n          <div class=\"kq-auth-panel\" data-panel=\"login\" style=\"display:none\">\n            <div class=\"kq-auth-card\">\n              <h3>Login<\/h3>\n              <div class=\"kq-field\">\n                <input id=\"kq-login-identity\" placeholder=\"Username or Email\" \/>\n              <\/div>\n              <div class=\"kq-field\">\n                <input id=\"kq-login-password\" placeholder=\"Password\" type=\"password\" \/>\n                <button class=\"kq-toggle-pass\" type=\"button\" aria-label=\"Toggle password\">\ud83d\udc41<\/button>\n              <\/div>\n              <button id=\"kq-login-btn\" class=\"button kq-btn-small\">Login<\/button>\n            <\/div>\n          <\/div>\n\n          <!-- FORGOT -->\n          <div class=\"kq-auth-panel\" data-panel=\"forgot\" style=\"display:none\">\n            <div class=\"kq-auth-card\">\n              <h3>Forgot Password<\/h3>\n              <div class=\"kq-field\">\n                <input id=\"kq-forgot-identity\" placeholder=\"Username or Email\" \/>\n              <\/div>\n              <div class=\"kq-field\">\n                <input id=\"kq-forgot-newpass\" placeholder=\"New Password\" type=\"password\" \/>\n                <button class=\"kq-toggle-pass\" type=\"button\" aria-label=\"Toggle password\">\ud83d\udc41<\/button>\n              <\/div>\n              <button id=\"kq-forgot-btn\" class=\"button kq-btn-small\">Update Password<\/button>\n            <\/div>\n          <\/div>\n\n        <\/div>\n\n      <\/div>\n    <\/div>\n    \n\n\n\n<div style=\"height:70px\" aria-hidden=\"true\" class=\"wp-block-spacer\"><\/div>\n\n\n\n<!-- HTML for the Desmos Calculator Embed (Always Visible) -->\n<div id=\"desmos-container\">\n    <iframe loading=\"lazy\"\n        src=\"https:\/\/www.desmos.com\/calculator\/fxgemyy2gl\"\n        width=\"100%\"\n        height=\"500px\"\n        frameborder=\"0\"\n        allowfullscreen\n    ><\/iframe>\n<\/div>\n\n<!-- Button to Open Calculator in Slide-Out Panel -->\n<button id=\"desmos-toggle\" style=\"position: fixed; top: 20px; right: 20px; z-index: 1000;\">\n    Open Calculator\n<\/button>\n\n<!-- Slide-Out Desmos Calculator Panel (hidden initially) -->\n<div id=\"desmos-panel\">\n    <iframe loading=\"lazy\"\n        src=\"https:\/\/www.desmos.com\/calculator\/fxgemyy2gl\"\n        width=\"100%\"\n        height=\"95%\"\n        frameborder=\"0\"\n        allowfullscreen\n    ><\/iframe>\n<\/div>\n\n<!-- CSS Styling for the Slide-Out Panel -->\n<style>\n    \/* Main Container Styling *\/\n    #desmos-container {\n        max-width: 600px; \/* Adjust as needed *\/\n        margin: 20px auto;\n    }\n\n    \/* Slide-Out Panel Styling *\/\n    #desmos-panel {\n        position: fixed;\n        top: 0;\n        right: -400px; \/* Hidden by default *\/\n        width: 400px; \/* Adjust width as needed *\/\n        height: 100vh;\n        background-color: white;\n        border-left: 1px solid #ccc;\n        box-shadow: -2px 0 5px rgba(0, 0, 0, 0.2);\n        transition: right 0.3s ease;\n        z-index: 999; \/* Ensure it overlays content *\/\n    }\n\n    #desmos-panel.open {\n        right: 0;\n    }\n<\/style>\n\n<!-- JavaScript to Toggle the Slide-Out Panel -->\n<script>\n    document.getElementById(\"desmos-toggle\").onclick = function() {\n        var panel = document.getElementById(\"desmos-panel\");\n        if (panel.classList.contains(\"open\")) {\n            panel.classList.remove(\"open\");\n        } else {\n            panel.classList.add(\"open\");\n        }\n    };\n<\/script>\n\n\n\n<p class=\"has-text-align-center has-small-font-size\">Wait for the Desmos Calculator to appear.<\/p>\n\n\n\n<div style=\"height:70px\" aria-hidden=\"true\" class=\"wp-block-spacer\"><\/div>\n\n\n\n<h3 class=\"wp-block-heading\">SAT MATH PROBLEM SOLUTIONS WITH STEP-BY-STEP EXPLANATION<\/h3>\n\n\n\n<p>Do not open the tabs before finishing the practice test above! For your convenience, we have compiled all the solutions and their explanations here. We will also give you some tips and advice to help you understand them better. You&#8217;ll see <strong>&#8216;why this answer is correct&#8217;<\/strong> and <strong>&#8216;why this is incorrect.&#8217;<\/strong><\/p>\n\n\n\n<div style=\"height:70px\" aria-hidden=\"true\" class=\"wp-block-spacer\"><\/div>\n\n\n\n<h4 class=\"wp-block-heading\">Math Solutions and Explanations:<\/h4>\n\n\n\n<p>The light red color shows the Question, green shows the Correct answer with step-by-step explanation, red shows the Incorrect one, and blue shows Desmos Tips or Tricks.<\/p>\n\n\n\n<div class=\"wp-block-coblocks-accordion alignfull\">\n<div class=\"wp-block-coblocks-accordion-item\"><details><summary class=\"wp-block-coblocks-accordion-item__title\"><strong>1st Question<\/strong><\/summary><div class=\"wp-block-coblocks-accordion-item__content\">\n<p class=\"is-style-warning\" style=\"font-size:0.9em\"><strong>Question:<\/strong> The combined original price for a mirror and a vase is $60. After a 25% discount to the mirror and a 45% discount to the vase are applied, the combined sale price for the two items is $39. Which system of equations gives the original price <math data-latex=\"m\"><semantics><mi>m<\/mi><annotation encoding=\"application\/x-tex\">m<\/annotation><\/semantics><\/math>, in dollars, of the mirror and the original price <math data-latex=\"n\"><semantics><mi>n<\/mi><annotation encoding=\"application\/x-tex\">n<\/annotation><\/semantics><\/math>, in dollars, of the vase?<br>A)<br><math><semantics><mrow><mi>m<\/mi><mo>+<\/mo><mi>v<\/mi><mo>=<\/mo><mn>60<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">m + v = 60<\/annotation><\/semantics><\/math><br><math><semantics><mrow><mn>0.55<\/mn><mi>m<\/mi><mo>+<\/mo><mn>0.75<\/mn><mi>v<\/mi><mo>=<\/mo><mn>39<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">0.55m + 0.75v = 39<\/annotation><\/semantics><\/math><br>B)<br><math><semantics><mrow><mi>m<\/mi><mo>+<\/mo><mi>v<\/mi><mo>=<\/mo><mn>60<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">m + v = 60<\/annotation><\/semantics><\/math><br><math><semantics><mrow><mn>0.45<\/mn><mi>m<\/mi><mo>+<\/mo><mn>0.25<\/mn><mi>v<\/mi><mo>=<\/mo><mn>39<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">0.45m + 0.25v = 39<\/annotation><\/semantics><\/math><br>C)<br><math><semantics><mrow><mi>m<\/mi><mo>+<\/mo><mi>v<\/mi><mo>=<\/mo><mn>60<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">m + v = 60<\/annotation><\/semantics><\/math><br><math><semantics><mrow><mn>0.75<\/mn><mi>m<\/mi><mo>+<\/mo><mn>0.55<\/mn><mi>v<\/mi><mo>=<\/mo><mn>39<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">0.75m + 0.55v = 39<\/annotation><\/semantics><\/math><br>D)<br><math><semantics><mrow><mi>m<\/mi><mo>+<\/mo><mi>v<\/mi><mo>=<\/mo><mn>60<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">m + v = 60<\/annotation><\/semantics><\/math><br><math><semantics><mrow><mn>0.25<\/mn><mi>m<\/mi><mo>+<\/mo><mn>0.45<\/mn><mi>v<\/mi><mo>=<\/mo><mn>39<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">0.25m + 0.45v = 39<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<p class=\"is-style-info\" style=\"font-size:0.9em\"><strong>Understand the Question<br><\/strong>~ Mirror + Vase original price = <strong>$60<\/strong><br>~ Mirror discount = <strong>25%<\/strong><br>~ Vase discount = <strong>45%<\/strong><br>~ Sale price total = <strong>$39<\/strong><br>Which system represents this?<br><br><strong>\ud83e\udde0 Core Concept (Discount logic)<br><\/strong>If an item has a <strong>25% discount<\/strong>, you pay:<br><math display=\"block\"><semantics><mrow><mn>100<\/mn><mi mathvariant=\"normal\">%<\/mi><mo>\u2212<\/mo><mn>25<\/mn><mi mathvariant=\"normal\">%<\/mi><mo>=<\/mo><mn>75<\/mn><mi mathvariant=\"normal\">%<\/mi><mo>=<\/mo><mn>0.75<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">100\\% &#8211; 25\\% = 75\\% = 0.75<\/annotation><\/semantics><\/math><br>If an item has a <strong>45% discount<\/strong>, you pay:<br><math display=\"block\"><semantics><mrow><mn>100<\/mn><mi mathvariant=\"normal\">%<\/mi><mo>\u2212<\/mo><mn>45<\/mn><mi mathvariant=\"normal\">%<\/mi><mo>=<\/mo><mn>55<\/mn><mi mathvariant=\"normal\">%<\/mi><mo>=<\/mo><mn>0.55<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">100\\% &#8211; 45\\% = 55\\% = 0.55<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<p class=\"is-style-success\" style=\"font-size:0.9em\"><strong>\ud83e\uddee Step-by-Step Explanation<br><\/strong>Original price equation:<br><math display=\"block\"><semantics><mrow><mi>m<\/mi><mo>+<\/mo><mi>v<\/mi><mo>=<\/mo><mn>60<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">m + v = 60<\/annotation><\/semantics><\/math><br>Sale price equation:<br><math data-latex=\"m\"><semantics><mi>m<\/mi><annotation encoding=\"application\/x-tex\">m<\/annotation><\/semantics><\/math> for mirror (75%<math data-latex=\"= \\frac{75}{100} = 0.75\"><semantics><mrow><mo>=<\/mo><mfrac><mn>75<\/mn><mn>100<\/mn><\/mfrac><mo>=<\/mo><mn>0.75<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">= \\frac{75}{100} = 0.75<\/annotation><\/semantics><\/math>)<br><br><math data-latex=\"v\"><semantics><mi>v<\/mi><annotation encoding=\"application\/x-tex\">v<\/annotation><\/semantics><\/math> for vase (55% <math data-latex=\"= \\frac{55}{100} = 0.55\"><semantics><mrow><mo>=<\/mo><mfrac><mn>55<\/mn><mn>100<\/mn><\/mfrac><mo>=<\/mo><mn>0.55<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">= \\frac{55}{100} = 0.55<\/annotation><\/semantics><\/math>)<br><math display=\"block\"><semantics><mrow><mn>0.75<\/mn><mi>m<\/mi><mo>+<\/mo><mn>0.55<\/mn><mi>v<\/mi><mo>=<\/mo><mn>39<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">0.75m + 0.55v = 39<\/annotation><\/semantics><\/math><br>\u2705 Correct Answer: <strong>C<\/strong><\/p>\n\n\n\n<p class=\"is-style-error\" style=\"font-size:0.9em\"><strong>Option A \u274c<\/strong><br><strong>Trap:<\/strong> Student subtracts discounts incorrectly.<\/p>\n\n\n\n<p class=\"is-style-error\" style=\"font-size:0.9em\"><strong>Option B \u274c<\/strong><br><strong>Trap:<\/strong> Student uses discount instead of remaining price.<\/p>\n\n\n\n<p class=\"is-style-error\" style=\"font-size:0.9em\"><strong>Option D \u274c<\/strong><br><strong>Trap:<\/strong> Student uses discount rates directly.<\/p>\n<\/div><\/details><\/div>\n\n\n\n<div class=\"wp-block-coblocks-accordion-item\"><details><summary class=\"wp-block-coblocks-accordion-item__title\"><strong>2nd Question<\/strong><\/summary><div class=\"wp-block-coblocks-accordion-item__content\">\n<figure class=\"wp-block-image size-full is-resized\"><img decoding=\"async\" src=\"https:\/\/us.mrenglishkj.com\/sat\/sat\/wp-content\/uploads\/2026\/01\/image_2026-01-11_204320296.png\" alt=\"Learn to solve graphs in SAT math for free - practice SAT exam\" class=\"wp-image-8682\" style=\"width:319px;height:auto\"\/><\/figure>\n\n\n\n<p class=\"is-style-warning\" style=\"font-size:0.9em\"><strong>Question:<\/strong> The graph of a system of an absolute value function and a linear function is shown. What is the solution <math data-latex=\"(x, y)\"><semantics><mrow><mo form=\"prefix\" stretchy=\"false\">(<\/mo><mi>x<\/mi><mo separator=\"true\">,<\/mo><mi>y<\/mi><mo form=\"postfix\" stretchy=\"false\">)<\/mo><\/mrow><annotation encoding=\"application\/x-tex\">(x, y)<\/annotation><\/semantics><\/math> to this system of two equations?<br>A) (-1, 5)<br>B) (0, 4)<br>C) (1, 5)<br>D) (4, 2)<\/p>\n\n\n\n<p class=\"is-style-info\"><strong>1\ufe0f\u20e3 Explain the Question<\/strong><br>~ A <strong>solution to a system<\/strong> is the <strong>intersection point<\/strong><br>~ The point must lie on <strong>both graphs<\/strong><br>~ You must read the <strong>exact intersection<\/strong> from the graph<br><br><strong>2\ufe0f\u20e3 Rules \/ Concepts Used<\/strong><br><strong>System of Equations (Graphical)<\/strong><br>~ Each solution = point where graphs cross<br>~ Only <strong>one intersection<\/strong> is shown<br>~ (x, y): It is absolute.<\/p>\n\n\n\n<p class=\"is-style-success\" style=\"font-size:0.9em\"><strong>3\ufe0f\u20e3 Step-by-Step Graph Analysis<\/strong><br>From the image:<br>~ The <strong>absolute value graph<\/strong> forms a \u201cV\u201d<br>~ The <strong>linear graph<\/strong> crosses it at exactly <strong>one point<\/strong><br>~ That intersection is clearly at: <math data-latex=\"(x, y)\"><semantics><mrow><mo form=\"prefix\" stretchy=\"false\">(<\/mo><mi>x<\/mi><mo separator=\"true\">,<\/mo><mi>y<\/mi><mo form=\"postfix\" stretchy=\"false\">)<\/mo><\/mrow><annotation encoding=\"application\/x-tex\">(x, y)<\/annotation><\/semantics><\/math><br><math><semantics><mrow><mi>x<\/mi><mo>=<\/mo><mn>1<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">x = 1<\/annotation><\/semantics><\/math><br><math><semantics><mrow><mi>y<\/mi><mo>=<\/mo><mn>5<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">y = 5<\/annotation><\/semantics><\/math><br><br>\u2705 <strong>Correct Answer: Option C<\/strong><\/p>\n\n\n\n<p class=\"is-style-error\" style=\"font-size:0.9em\">\u274c <strong>(-1, 5)<\/strong> \u2192 Lies on one graph, not both<br>\u274c <strong>(0, 4)<\/strong> \u2192 On the line but not on the V<br>\u274c <strong>(4, 2)<\/strong> \u2192 Vertex of V but no line there<\/p>\n<\/div><\/details><\/div>\n\n\n\n<div class=\"wp-block-coblocks-accordion-item\"><details><summary class=\"wp-block-coblocks-accordion-item__title\"><strong>3rd Question<\/strong><\/summary><div class=\"wp-block-coblocks-accordion-item__content\">\n<p class=\"is-style-warning\" style=\"font-size:0.9em\"><strong>Question:<\/strong><br><math data-latex=\"4a^2 + 20ab + 25b^2\"><semantics><mrow><mn>4<\/mn><msup><mi>a<\/mi><mn>2<\/mn><\/msup><mo>+<\/mo><mn>20<\/mn><mi>a<\/mi><mi>b<\/mi><mo>+<\/mo><mn>25<\/mn><msup><mi>b<\/mi><mn>2<\/mn><\/msup><\/mrow><annotation encoding=\"application\/x-tex\">4a^2 + 20ab + 25b^2<\/annotation><\/semantics><\/math><br>Which of the following is a factor of the polynomial above?<br>A) <math data-latex=\"a + b\"><semantics><mrow><mi>a<\/mi><mo>+<\/mo><mi>b<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">a + b<\/annotation><\/semantics><\/math><br>B) <math data-latex=\"2a +5b\"><semantics><mrow><mn>2<\/mn><mi>a<\/mi><mo>+<\/mo><mn>5<\/mn><mi>b<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">2a +5b<\/annotation><\/semantics><\/math><br>C) <math data-latex=\"4a + 5b\"><semantics><mrow><mn>4<\/mn><mi>a<\/mi><mo>+<\/mo><mn>5<\/mn><mi>b<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">4a + 5b<\/annotation><\/semantics><\/math><br>D) <math data-latex=\"4a + 25b\"><semantics><mrow><mn>4<\/mn><mi>a<\/mi><mo>+<\/mo><mn>25<\/mn><mi>b<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">4a + 25b<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<p class=\"is-style-info\" style=\"font-size:0.9em\"><strong>\ud83e\udde0 Core Concept (Recognize structure first)<\/strong><br>This matches:<math display=\"block\"><semantics><mrow><mo stretchy=\"false\">(<\/mo><mi>A<\/mi><mo>+<\/mo><mi>B<\/mi><msup><mo stretchy=\"false\">)<\/mo><mn>2<\/mn><\/msup><mo>=<\/mo><msup><mi>A<\/mi><mn>2<\/mn><\/msup><mo>+<\/mo><mn>2<\/mn><mi>A<\/mi><mi>B<\/mi><mo>+<\/mo><msup><mi>B<\/mi><mn>2<\/mn><\/msup><\/mrow><annotation encoding=\"application\/x-tex\">(A + B)^2 = A^2 + 2AB + B^2<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<p class=\"is-style-success\" style=\"font-size:0.9em\"><strong>\ud83e\uddee Step-by-Step Solution<\/strong><br><strong>Step 1: Arrange into Formula<\/strong><br>We have: <math data-latex=\"4a^2 + 20ab + 25b^2\"><semantics><mrow><mn>4<\/mn><msup><mi>a<\/mi><mn>2<\/mn><\/msup><mo>+<\/mo><mn>20<\/mn><mi>a<\/mi><mi>b<\/mi><mo>+<\/mo><mn>25<\/mn><msup><mi>b<\/mi><mn>2<\/mn><\/msup><\/mrow><annotation encoding=\"application\/x-tex\">4a^2 + 20ab + 25b^2<\/annotation><\/semantics><\/math><br><math data-latex=\"A^2 = 4a^2\"><semantics><mrow><msup><mi>A<\/mi><mn>2<\/mn><\/msup><mo>=<\/mo><mn>4<\/mn><msup><mi>a<\/mi><mn>2<\/mn><\/msup><\/mrow><annotation encoding=\"application\/x-tex\">A^2 = 4a^2<\/annotation><\/semantics><\/math>    then what is   <math data-latex=\"A = 2a\"><semantics><mrow><mi>A<\/mi><mo>=<\/mo><mn>2<\/mn><mi>a<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">A = 2a<\/annotation><\/semantics><\/math><br><math data-latex=\"B^2 = 25b^2\"><semantics><mrow><msup><mi>B<\/mi><mn>2<\/mn><\/msup><mo>=<\/mo><mn>25<\/mn><msup><mi>b<\/mi><mn>2<\/mn><\/msup><\/mrow><annotation encoding=\"application\/x-tex\">B^2 = 25b^2<\/annotation><\/semantics><\/math>    then what is   <math data-latex=\"B = 5b\"><semantics><mrow><mi>B<\/mi><mo>=<\/mo><mn>5<\/mn><mi>b<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">B = 5b<\/annotation><\/semantics><\/math><br><math data-latex=\"2AB = 20ab\"><semantics><mrow><mn>2<\/mn><mi>A<\/mi><mi>B<\/mi><mo>=<\/mo><mn>20<\/mn><mi>a<\/mi><mi>b<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">2AB = 20ab<\/annotation><\/semantics><\/math>    then its expanded form is   <math data-latex=\"2(A)(B) = 2(2a)(5b)\"><semantics><mrow><mn>2<\/mn><mo form=\"prefix\" stretchy=\"false\">(<\/mo><mi>A<\/mi><mo form=\"postfix\" stretchy=\"false\">)<\/mo><mo form=\"prefix\" stretchy=\"false\">(<\/mo><mi>B<\/mi><mo form=\"postfix\" stretchy=\"false\">)<\/mo><mo>=<\/mo><mn>2<\/mn><mo form=\"prefix\" stretchy=\"false\">(<\/mo><mn>2<\/mn><mi>a<\/mi><mo form=\"postfix\" stretchy=\"false\">)<\/mo><mo form=\"prefix\" stretchy=\"false\">(<\/mo><mn>5<\/mn><mi>b<\/mi><mo form=\"postfix\" stretchy=\"false\">)<\/mo><\/mrow><annotation encoding=\"application\/x-tex\">2(A)(B) = 2(2a)(5b)<\/annotation><\/semantics><\/math><br><br>Rewrite:<br><math data-latex=\"4a^2 + 20ab + 25b^2\\\\(2a)^2 + 2(2a)(5b) + (5b)^2\\\\(2a + 5b)^2\"><semantics><mtable columnalign=\"left\" rowspacing=\"0em\"><mtr><mtd style=\"text-align:left\"><mrow><mn>4<\/mn><msup><mi>a<\/mi><mn>2<\/mn><\/msup><mo>+<\/mo><mn>20<\/mn><mi>a<\/mi><mi>b<\/mi><mo>+<\/mo><mn>25<\/mn><msup><mi>b<\/mi><mn>2<\/mn><\/msup><\/mrow><mo><\/mo><\/mtd><\/mtr><mtr><mtd style=\"text-align:left\"><mrow><mo form=\"prefix\" stretchy=\"false\">(<\/mo><mn>2<\/mn><mi>a<\/mi><msup><mo form=\"postfix\" stretchy=\"false\">)<\/mo><mn>2<\/mn><\/msup><mo>+<\/mo><mn>2<\/mn><mo form=\"prefix\" stretchy=\"false\">(<\/mo><mn>2<\/mn><mi>a<\/mi><mo form=\"postfix\" stretchy=\"false\">)<\/mo><mo form=\"prefix\" stretchy=\"false\">(<\/mo><mn>5<\/mn><mi>b<\/mi><mo form=\"postfix\" stretchy=\"false\">)<\/mo><mo>+<\/mo><mo form=\"prefix\" stretchy=\"false\">(<\/mo><mn>5<\/mn><mi>b<\/mi><msup><mo form=\"postfix\" stretchy=\"false\">)<\/mo><mn>2<\/mn><\/msup><\/mrow><mo><\/mo><\/mtd><\/mtr><mtr><mtd style=\"text-align:left\"><mrow><mo form=\"prefix\" stretchy=\"false\">(<\/mo><mn>2<\/mn><mi>a<\/mi><mo>+<\/mo><mn>5<\/mn><mi>b<\/mi><msup><mo form=\"postfix\" stretchy=\"false\">)<\/mo><mn>2<\/mn><\/msup><\/mrow><\/mtd><\/mtr><\/mtable><annotation encoding=\"application\/x-tex\">4a^2 + 20ab + 25b^2\\\\(2a)^2 + 2(2a)(5b) + (5b)^2\\\\(2a + 5b)^2<\/annotation><\/semantics><\/math><br><br><strong>\u2705 Correct Answer: B<\/strong><\/p>\n\n\n\n<p class=\"is-style-error\" style=\"font-size:0.9em\"><strong>a + b \u274c<\/strong><br>Trap: Student ignores coefficients.<\/p>\n\n\n\n<p class=\"is-style-error\" style=\"font-size:0.9em\"><strong>4a + 5b \u274c<\/strong><br>Trap: Student partially factors.<\/p>\n\n\n\n<p class=\"is-style-error\" style=\"font-size:0.9em\"><strong>4a + 5b \u274c<\/strong><br>Trap: Student partially factors.<\/p>\n\n\n\n<p class=\"is-style-info\" style=\"font-size:0.9em\"><strong>\ud83e\uddee DESMOS <\/strong><br>1. Type: factor(4a^2 + 20ab + 25b^2)<br>2. Output: (2a + 5b)^2<br><math data-latex=\"(2a + 5b)^2\"><semantics><mrow><mo form=\"prefix\" stretchy=\"false\">(<\/mo><mn>2<\/mn><mi>a<\/mi><mo>+<\/mo><mn>5<\/mn><mi>b<\/mi><msup><mo form=\"postfix\" stretchy=\"false\">)<\/mo><mn>2<\/mn><\/msup><\/mrow><annotation encoding=\"application\/x-tex\">(2a + 5b)^2<\/annotation><\/semantics><\/math><\/p>\n<\/div><\/details><\/div>\n\n\n\n<div class=\"wp-block-coblocks-accordion-item\"><details><summary class=\"wp-block-coblocks-accordion-item__title\"><strong>4th Question<\/strong><\/summary><div class=\"wp-block-coblocks-accordion-item__content\">\n<figure class=\"wp-block-table\"><table class=\"has-fixed-layout\"><thead><tr><th class=\"has-text-align-center\" data-align=\"center\">Data Value<\/th><th class=\"has-text-align-center\" data-align=\"center\">Frequency<\/th><\/tr><\/thead><tbody><tr><td class=\"has-text-align-center\" data-align=\"center\">6<\/td><td class=\"has-text-align-center\" data-align=\"center\">3<\/td><\/tr><tr><td class=\"has-text-align-center\" data-align=\"center\">7<\/td><td class=\"has-text-align-center\" data-align=\"center\">3<\/td><\/tr><tr><td class=\"has-text-align-center\" data-align=\"center\">8<\/td><td class=\"has-text-align-center\" data-align=\"center\">8<\/td><\/tr><tr><td class=\"has-text-align-center\" data-align=\"center\">9<\/td><td class=\"has-text-align-center\" data-align=\"center\">8<\/td><\/tr><tr><td class=\"has-text-align-center\" data-align=\"center\">10<\/td><td class=\"has-text-align-center\" data-align=\"center\">9<\/td><\/tr><tr><td class=\"has-text-align-center\" data-align=\"center\">11<\/td><td class=\"has-text-align-center\" data-align=\"center\">16<\/td><\/tr><tr><td class=\"has-text-align-center\" data-align=\"center\">12<\/td><td class=\"has-text-align-center\" data-align=\"center\">9<\/td><\/tr><tr><td class=\"has-text-align-center\" data-align=\"center\">13<\/td><td class=\"has-text-align-center\" data-align=\"center\">0<\/td><\/tr><tr><td class=\"has-text-align-center\" data-align=\"center\">14<\/td><td class=\"has-text-align-center\" data-align=\"center\">6<\/td><\/tr><\/tbody><\/table><\/figure>\n\n\n\n<p class=\"is-style-warning\" style=\"font-size:0.9em\"><strong>Question:<\/strong> The frequency table summarizes the 57 data values in a data set. What is the maximum data value in the data set?<br>A) 11<br>B) 12<br>C) 14<br>D) 16<\/p>\n\n\n\n<p class=\"is-style-info\" style=\"font-size:0.9em\"><strong>Important Rule<\/strong><br>The <strong>maximum data value<\/strong> is:<br>The <strong>largest data value with a nonzero frequency<\/strong><\/p>\n\n\n\n<p class=\"is-style-success\" style=\"font-size:0.9em\"><strong>Step-by-Step Solution<\/strong><\/p>\n\n\n\n<figure class=\"wp-block-table\"><table class=\"has-fixed-layout\"><thead><tr><th class=\"has-text-align-center\" data-align=\"center\">Data Value<\/th><th class=\"has-text-align-center\" data-align=\"center\">Frequency<\/th><\/tr><\/thead><tbody><tr><td class=\"has-text-align-center\" data-align=\"center\">6<\/td><td class=\"has-text-align-center\" data-align=\"center\">3<\/td><\/tr><tr><td class=\"has-text-align-center\" data-align=\"center\">7<\/td><td class=\"has-text-align-center\" data-align=\"center\">3<\/td><\/tr><tr><td class=\"has-text-align-center\" data-align=\"center\">8<\/td><td class=\"has-text-align-center\" data-align=\"center\">8<\/td><\/tr><tr><td class=\"has-text-align-center\" data-align=\"center\">9<\/td><td class=\"has-text-align-center\" data-align=\"center\">8<\/td><\/tr><tr><td class=\"has-text-align-center\" data-align=\"center\">10<\/td><td class=\"has-text-align-center\" data-align=\"center\">9<\/td><\/tr><tr><td class=\"has-text-align-center\" data-align=\"center\">11<\/td><td class=\"has-text-align-center\" data-align=\"center\">16<\/td><\/tr><tr><td class=\"has-text-align-center\" data-align=\"center\">12<\/td><td class=\"has-text-align-center\" data-align=\"center\">9<\/td><\/tr><tr><td class=\"has-text-align-center\" data-align=\"center\">13<\/td><td class=\"has-text-align-center\" data-align=\"center\"><strong>0<\/strong> \u274c<\/td><\/tr><tr><td class=\"has-text-align-center\" data-align=\"center\">14<\/td><td class=\"has-text-align-center\" data-align=\"center\"><strong>6<\/strong> \u2705<\/td><\/tr><\/tbody><\/table><\/figure>\n\n\n\n<p class=\"is-style-success\">We just need to know the Largest Data Value, we do not need to multiply Data Value to its Frequency to find out the largest. Just choose the Largest Data Value.<br>Data value <strong>14<\/strong> has frequency <strong>6<\/strong><br><strong>Correct Answer<\/strong><math display=\"block\"><semantics><mrow><menclose notation=\"box\"><mstyle scriptlevel=\"0\" displaystyle=\"false\"><mstyle scriptlevel=\"0\" displaystyle=\"false\"><mstyle scriptlevel=\"0\" displaystyle=\"true\"><mn>14<\/mn><\/mstyle><\/mstyle><\/mstyle><\/menclose><\/mrow><annotation encoding=\"application\/x-tex\">\\boxed{14}<\/annotation><\/semantics><\/math><br>\u2705 <strong>Option: 14<\/strong><\/p>\n\n\n\n<p class=\"is-style-error\"><strong>Why Other Options Are Incorrect<\/strong><br><strong>11<\/strong> \u274c Not the largest value<br><strong>12<\/strong> \u274c Still smaller than 14<br><strong>16<\/strong> \u274c This is a frequency, not a data value<br><br><strong>Common Student Mistakes<\/strong><br>\u274c Confusing \u201chow many\u201d with \u201cwhat value\u201d<br>\u274c Choosing the largest frequency<br>\u274c Ignoring zero frequency<\/p>\n<\/div><\/details><\/div>\n\n\n\n<div class=\"wp-block-coblocks-accordion-item\"><details><summary class=\"wp-block-coblocks-accordion-item__title\"><strong>5th Question<\/strong><\/summary><div class=\"wp-block-coblocks-accordion-item__content\">\n<p class=\"is-style-warning\" style=\"font-size:0.9em\"><strong>Question:<\/strong> The measure of angle <math data-latex=\"R\"><semantics><mi>R<\/mi><annotation encoding=\"application\/x-tex\">R<\/annotation><\/semantics><\/math> is <math data-latex=\"\\frac{2 \\pi}{3}\"><semantics><mfrac><mrow><mn>2<\/mn><mi>\u03c0<\/mi><\/mrow><mn>3<\/mn><\/mfrac><annotation encoding=\"application\/x-tex\">\\frac{2 \\pi}{3}<\/annotation><\/semantics><\/math> radians. The measure of angle <math data-latex=\"T\"><semantics><mi>T<\/mi><annotation encoding=\"application\/x-tex\">T<\/annotation><\/semantics><\/math> is <math data-latex=\"\\frac{5 \\pi}{12}\"><semantics><mfrac><mrow><mn>5<\/mn><mi>\u03c0<\/mi><\/mrow><mn>12<\/mn><\/mfrac><annotation encoding=\"application\/x-tex\">\\frac{5 \\pi}{12}<\/annotation><\/semantics><\/math> radians greater than the measure of angle <math data-latex=\"R\"><semantics><mi>R<\/mi><annotation encoding=\"application\/x-tex\">R<\/annotation><\/semantics><\/math>. What is the measure of angle <math data-latex=\"T\"><semantics><mi>T<\/mi><annotation encoding=\"application\/x-tex\">T<\/annotation><\/semantics><\/math>, in degrees?<br>A) 75<br>B) 120<br>C) 195<br>D) 390<\/p>\n\n\n\n<p class=\"is-style-info\" style=\"font-size:0.9em\"><strong>Understanding the Question<\/strong><br>You are given:<br>Angle R in <strong>radians<\/strong><br>Angle T is <strong>greater than R by<\/strong> a given radian amount<br>You must convert the final angle into <strong>degrees<\/strong>.<br><br><strong>Important Rules<\/strong><br><strong>Add radians first<\/strong><br>Convert radians \u2192 degrees using:<math display=\"block\"><semantics><mrow><mtext>Degrees<\/mtext><mo>=<\/mo><mtext>Radians<\/mtext><mo>\u00d7<\/mo><mfrac><msup><mn>180<\/mn><mo>\u2218<\/mo><\/msup><mi>\u03c0<\/mi><\/mfrac><\/mrow><annotation encoding=\"application\/x-tex\">\\text{Degrees} = \\text{Radians} \\times \\frac{180^\\circ}{\\pi}<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<p class=\"is-style-success\" style=\"font-size:0.9em\"><strong>Step-by-Step Solution<\/strong><br><strong>Step 1: Given Values<\/strong><math display=\"block\"><semantics><mrow><mi>R<\/mi><mo>=<\/mo><mfrac><mrow><mn>2<\/mn><mi>\u03c0<\/mi><\/mrow><mn>3<\/mn><\/mfrac><\/mrow><annotation encoding=\"application\/x-tex\">R = \\frac{2\\pi}{3}<\/annotation><\/semantics><\/math><strong>T<\/strong> is greater than <strong>R<\/strong>, so<br><math display=\"block\"><semantics><mrow><mi>T<\/mi><mo>=<\/mo><mi>R<\/mi><mo>+<\/mo><mfrac><mrow><mn>5<\/mn><mi>\u03c0<\/mi><\/mrow><mn>12<\/mn><\/mfrac><\/mrow><annotation encoding=\"application\/x-tex\">T = R + \\frac{5\\pi}{12}<\/annotation><\/semantics><\/math><br><strong>Step 2: Add the Radians<\/strong><br>Find a common denominator between 3 and 12 LCM:<br><math data-latex=\"\\frac{2 \\pi}{3} + \\frac{5 \\pi}{12} \\\\ \\\\ \\frac{2 \\pi \\times 4 + 5 \\pi \\times 1}{12} \\\\ \\\\ \\frac{8 \\pi + 5 \\pi}{12} \\\\ \\\\ \\frac{13 \\pi}{12}\"><semantics><mtable columnalign=\"left\" rowspacing=\"0em\"><mtr><mtd style=\"text-align:left\"><mrow><mfrac><mrow><mn>2<\/mn><mi>\u03c0<\/mi><\/mrow><mn>3<\/mn><\/mfrac><mo>+<\/mo><mfrac><mrow><mn>5<\/mn><mi>\u03c0<\/mi><\/mrow><mn>12<\/mn><\/mfrac><\/mrow><mo><\/mo><\/mtd><\/mtr><mtr><mtd style=\"text-align:left\"><mo><\/mo><\/mtd><\/mtr><mtr><mtd style=\"text-align:left\"><mrow><mfrac><mrow><mn>2<\/mn><mi>\u03c0<\/mi><mo>\u00d7<\/mo><mn>4<\/mn><mo>+<\/mo><mn>5<\/mn><mi>\u03c0<\/mi><mo>\u00d7<\/mo><mn>1<\/mn><\/mrow><mn>12<\/mn><\/mfrac><\/mrow><mo><\/mo><\/mtd><\/mtr><mtr><mtd style=\"text-align:left\"><mo><\/mo><\/mtd><\/mtr><mtr><mtd style=\"text-align:left\"><mrow><mfrac><mrow><mn>8<\/mn><mi>\u03c0<\/mi><mo>+<\/mo><mn>5<\/mn><mi>\u03c0<\/mi><\/mrow><mn>12<\/mn><\/mfrac><\/mrow><mo><\/mo><\/mtd><\/mtr><mtr><mtd style=\"text-align:left\"><mo><\/mo><\/mtd><\/mtr><mtr><mtd style=\"text-align:left\"><mrow><mfrac><mrow><mn>13<\/mn><mi>\u03c0<\/mi><\/mrow><mn>12<\/mn><\/mfrac><\/mrow><\/mtd><\/mtr><\/mtable><annotation encoding=\"application\/x-tex\">\\frac{2 \\pi}{3} + \\frac{5 \\pi}{12} \\\\ \\\\ \\frac{2 \\pi \\times 4 + 5 \\pi \\times 1}{12} \\\\ \\\\ \\frac{8 \\pi + 5 \\pi}{12} \\\\ \\\\ \\frac{13 \\pi}{12}<\/annotation><\/semantics><\/math><br><br><strong>Step 3: Convert to Degrees<\/strong><math display=\"block\"><semantics><mrow><mfrac><mrow><mn>13<\/mn><mi>\u03c0<\/mi><\/mrow><mn>12<\/mn><\/mfrac><mo>\u00d7<\/mo><mfrac><mn>180<\/mn><mi>\u03c0<\/mi><\/mfrac><mo>=<\/mo><mfrac><mrow><mn>13<\/mn><mo>\u00d7<\/mo><mn>180<\/mn><\/mrow><mn>12<\/mn><\/mfrac><\/mrow><annotation encoding=\"application\/x-tex\">\\frac{13\\pi}{12} \\times \\frac{180}{\\pi} = \\frac{13 \\times 180}{12}<\/annotation><\/semantics><\/math>Divide 180 by 12<br><math display=\"block\"><semantics><mrow><mo>=<\/mo><mn>13<\/mn><mo>\u00d7<\/mo><mn>15<\/mn><mo>=<\/mo><msup><mn>195<\/mn><mo>\u2218<\/mo><\/msup><\/mrow><annotation encoding=\"application\/x-tex\">= 13 \\times 15 = 195^\\circ<\/annotation><\/semantics><\/math><br><strong>Correct Answer: Option C<\/strong><math display=\"block\"><semantics><mrow><menclose notation=\"box\"><mstyle scriptlevel=\"0\" displaystyle=\"false\"><mstyle scriptlevel=\"0\" displaystyle=\"false\"><mstyle scriptlevel=\"0\" displaystyle=\"true\"><msup><mn>195<\/mn><mo>\u2218<\/mo><\/msup><\/mstyle><\/mstyle><\/mstyle><\/menclose><\/mrow><annotation encoding=\"application\/x-tex\">\\boxed{195^\\circ}<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<p class=\"is-style-error\" style=\"font-size:0.9em\"><strong>Why Other Options Are Incorrect<\/strong><br><strong>75\u00b0<\/strong> \u274c Too small, ignores addition<br><strong>120\u00b0<\/strong> \u274c Equals <math><semantics><mrow><mfrac><mrow><mn>2<\/mn><mi>\u03c0<\/mi><\/mrow><mn>3<\/mn><\/mfrac><\/mrow><annotation encoding=\"application\/x-tex\">\\frac{2\\pi}{3}<\/annotation><\/semantics><\/math>, not T<br><strong>390\u00b0<\/strong> \u274c Arithmetic error<br><br><strong>Common Student Mistakes<\/strong><br>Dropping \u03c0 incorrectly<br>Converting before adding<br>Forgetting common denominators<\/p>\n\n\n\n<p class=\"is-style-info\" style=\"font-size:0.9em\">Use DESMOS as a calculator only.<\/p>\n<\/div><\/details><\/div>\n\n\n\n<div class=\"wp-block-coblocks-accordion-item\"><details><summary class=\"wp-block-coblocks-accordion-item__title\"><strong>6th Question<\/strong><\/summary><div class=\"wp-block-coblocks-accordion-item__content\">\n<p class=\"is-style-warning\" style=\"font-size:0.9em\"><strong>Question:<\/strong><br><math data-latex=\"a^2\u2212a\u22121=0\"><semantics><mrow><msup><mi>a<\/mi><mn>2<\/mn><\/msup><mo>\u2212<\/mo><mi>a<\/mi><mo>\u2212<\/mo><mn>1<\/mn><mo>=<\/mo><mn>0<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">a^2\u2212a\u22121=0<\/annotation><\/semantics><\/math><br>What values satisfy the equation?<br>A) <math><semantics><mrow><mi>a<\/mi><mo>=<\/mo><mn>1<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">a = 1<\/annotation><\/semantics><\/math> and <math><semantics><mrow><mi>a<\/mi><mo>=<\/mo><mn>2<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">a = 2<\/annotation><\/semantics><\/math><br>B) <math><semantics><mrow><mi>a<\/mi><mo>=<\/mo><mo>\u2212<\/mo><mfrac><mn>1<\/mn><mn>2<\/mn><\/mfrac><\/mrow><annotation encoding=\"application\/x-tex\">a = -\\frac12<\/annotation><\/semantics><\/math>\u200b and <math><semantics><mrow><mi>a<\/mi><mo>=<\/mo><mfrac><mn>3<\/mn><mn>2<\/mn><\/mfrac><\/mrow><annotation encoding=\"application\/x-tex\">a = \\frac32<\/annotation><\/semantics><\/math>\u200b<br>C) <math><semantics><mrow><mi>a<\/mi><mo>=<\/mo><mfrac><mrow><mn>1<\/mn><mo>+<\/mo><msqrt><mn>5<\/mn><\/msqrt><\/mrow><mn>2<\/mn><\/mfrac><\/mrow><annotation encoding=\"application\/x-tex\">a = \\frac{1 + \\sqrt{5}}{2}<\/annotation><\/semantics><\/math> and <math><semantics><mrow><mi>a<\/mi><mo>=<\/mo><mfrac><mrow><mn>1<\/mn><mo>\u2212<\/mo><msqrt><mn>5<\/mn><\/msqrt><\/mrow><mn>2<\/mn><\/mfrac><\/mrow><annotation encoding=\"application\/x-tex\">a = \\frac{1 &#8211; \\sqrt{5}}{2}<\/annotation><\/semantics><\/math><br>D) <math><semantics><mrow><mi>a<\/mi><mo>=<\/mo><mfrac><mrow><mo>\u2212<\/mo><mn>1<\/mn><mo>+<\/mo><msqrt><mn>5<\/mn><\/msqrt><\/mrow><mn>2<\/mn><\/mfrac><\/mrow><annotation encoding=\"application\/x-tex\">a = \\frac{-1 + \\sqrt{5}}{2}<\/annotation><\/semantics><\/math>\u200b\u200b and <math><semantics><mrow><mi>a<\/mi><mo>=<\/mo><mfrac><mrow><mo>\u2212<\/mo><mn>1<\/mn><mo>\u2212<\/mo><msqrt><mn>5<\/mn><\/msqrt><\/mrow><mn>2<\/mn><\/mfrac><\/mrow><annotation encoding=\"application\/x-tex\">a = \\frac{-1 &#8211; \\sqrt{5}}{2}<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<p class=\"is-style-info\" style=\"font-size:0.9em\"><strong>\ud83e\udde0 Core Concept (Why quadratic formula is needed)<\/strong><br>This quadratic:<math display=\"block\"><semantics><mrow><msup><mi>a<\/mi><mn>2<\/mn><\/msup><mo>\u2212<\/mo><mi>a<\/mi><mo>\u2212<\/mo><mn>1<\/mn><mo>=<\/mo><mn>0<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">a^2 &#8211; a &#8211; 1 = 0<\/annotation><\/semantics><\/math><br>~ does <strong>not factor nicely<\/strong><br>~ coefficients are small but roots involve <strong>irrational numbers<\/strong><br>So the <strong>quadratic formula<\/strong> is the correct and safest method.<\/p>\n\n\n\n<p class=\"is-style-success\" style=\"font-size:0.9em\"><strong>\ud83e\uddee Step-by-Step Solution<br><\/strong>Quadratic formula:<math display=\"block\"><semantics><mrow><mi>a<\/mi><mo>=<\/mo><mfrac><mrow><mo>\u2212<\/mo><mi>b<\/mi><mo>\u00b1<\/mo><msqrt><mrow><msup><mi>b<\/mi><mn>2<\/mn><\/msup><mo>\u2212<\/mo><mn>4<\/mn><mi>a<\/mi><mi>c<\/mi><\/mrow><\/msqrt><\/mrow><mrow><mn>2<\/mn><mi>a<\/mi><\/mrow><\/mfrac><\/mrow><annotation encoding=\"application\/x-tex\">a = \\frac{-b \\pm \\sqrt{b^2 &#8211; 4ac}}{2a}<\/annotation><\/semantics><\/math><br>Identify coefficients:<br><math><semantics><mrow><mi>a<\/mi><mo>=<\/mo><mn>1<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">a = 1<\/annotation><\/semantics><\/math><br><math><semantics><mrow><mi>b<\/mi><mo>=<\/mo><mo>\u2212<\/mo><mn>1<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">b = -1<\/annotation><\/semantics><\/math><br><math><semantics><mrow><mi>c<\/mi><mo>=<\/mo><mo>\u2212<\/mo><mn>1<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">c = -1<\/annotation><\/semantics><\/math><br><br><strong>Step 1: Compute the discriminant<\/strong><br><math display=\"block\"><semantics><mrow><msup><mi>b<\/mi><mn>2<\/mn><\/msup><mo>\u2212<\/mo><mn>4<\/mn><mi>a<\/mi><mi>c<\/mi><mo>=<\/mo><mo stretchy=\"false\">(<\/mo><mo>\u2212<\/mo><mn>1<\/mn><msup><mo stretchy=\"false\">)<\/mo><mn>2<\/mn><\/msup><mo>\u2212<\/mo><mn>4<\/mn><mo stretchy=\"false\">(<\/mo><mn>1<\/mn><mo stretchy=\"false\">)<\/mo><mo stretchy=\"false\">(<\/mo><mo>\u2212<\/mo><mn>1<\/mn><mo stretchy=\"false\">)<\/mo><\/mrow><annotation encoding=\"application\/x-tex\">b^2 &#8211; 4ac = (-1)^2 &#8211; 4(1)(-1)<\/annotation><\/semantics><\/math><br>~ Minus, Minus = Plus, so <math data-latex=\"-1^2 = -1 \\times -1 = 1\"><semantics><mrow><mo>\u2212<\/mo><msup><mn>1<\/mn><mn>2<\/mn><\/msup><mo>=<\/mo><mo form=\"prefix\" stretchy=\"false\">\u2212<\/mo><mn>1<\/mn><mo>\u00d7<\/mo><mo form=\"prefix\" stretchy=\"false\">\u2212<\/mo><mn>1<\/mn><mo>=<\/mo><mn>1<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">-1^2 = -1 \\times -1 = 1<\/annotation><\/semantics><\/math><br>~ Same here = <math data-latex=\"-4 \\times 1 \\times -1 = -4 \\times -1 = 4\"><semantics><mrow><mo>\u2212<\/mo><mn>4<\/mn><mo>\u00d7<\/mo><mn>1<\/mn><mo>\u00d7<\/mo><mo form=\"prefix\" stretchy=\"false\">\u2212<\/mo><mn>1<\/mn><mo>=<\/mo><mo form=\"prefix\" stretchy=\"false\">\u2212<\/mo><mn>4<\/mn><mo>\u00d7<\/mo><mo form=\"prefix\" stretchy=\"false\">\u2212<\/mo><mn>1<\/mn><mo>=<\/mo><mn>4<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">-4 \\times 1 \\times -1 = -4 \\times -1 = 4<\/annotation><\/semantics><\/math><br><math display=\"block\"><semantics><mrow><mo>=<\/mo><mn>1<\/mn><mo>+<\/mo><mn>4<\/mn><mo>=<\/mo><mn>5<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">= 1 + 4 = 5<\/annotation><\/semantics><\/math><br><strong>Step 2: Substitute into the formula<\/strong><br><br><math data-latex=\"a = \\frac{-b \\pm \\sqrt{b^2 - 4ac}}{2a}\"><semantics><mrow><mi>a<\/mi><mo>=<\/mo><mfrac><mrow><mo lspace=\"0em\" rspace=\"0em\">\u2212<\/mo><mi>b<\/mi><mo>\u00b1<\/mo><msqrt><mrow><msup><mi>b<\/mi><mn>2<\/mn><\/msup><mo>\u2212<\/mo><mn>4<\/mn><mi>a<\/mi><mi>c<\/mi><\/mrow><\/msqrt><\/mrow><mrow><mn>2<\/mn><mi>a<\/mi><\/mrow><\/mfrac><\/mrow><annotation encoding=\"application\/x-tex\">a = \\frac{-b \\pm \\sqrt{b^2 &#8211; 4ac}}{2a}<\/annotation><\/semantics><\/math><br><br>Let&#8217;s input the already calculate part:<br><br><math data-latex=\"a = \\frac{-b \\pm \\sqrt{5}}{2a}\"><semantics><mrow><mi>a<\/mi><mo>=<\/mo><mfrac><mrow><mo lspace=\"0em\" rspace=\"0em\">\u2212<\/mo><mi>b<\/mi><mo>\u00b1<\/mo><msqrt><mn>5<\/mn><\/msqrt><\/mrow><mrow><mn>2<\/mn><mi>a<\/mi><\/mrow><\/mfrac><\/mrow><annotation encoding=\"application\/x-tex\">a = \\frac{-b \\pm \\sqrt{5}}{2a}<\/annotation><\/semantics><\/math><br><br>Now, let&#8217;s input other values:<br><br><math data-latex=\"a = \\frac{-(-1) \\pm \\sqrt{5}}{(2)(1)}\"><semantics><mrow><mi>a<\/mi><mo>=<\/mo><mfrac><mrow><mo lspace=\"0em\" rspace=\"0em\">\u2212<\/mo><mo form=\"prefix\" stretchy=\"false\">(<\/mo><mo form=\"prefix\" stretchy=\"false\">\u2212<\/mo><mn>1<\/mn><mo form=\"postfix\" stretchy=\"false\">)<\/mo><mo>\u00b1<\/mo><msqrt><mn>5<\/mn><\/msqrt><\/mrow><mrow><mo form=\"prefix\" stretchy=\"false\" lspace=\"0em\" rspace=\"0em\">(<\/mo><mn>2<\/mn><mo form=\"postfix\" stretchy=\"false\">)<\/mo><mo form=\"prefix\" stretchy=\"false\">(<\/mo><mn>1<\/mn><mo form=\"postfix\" stretchy=\"false\" lspace=\"0em\" rspace=\"0em\">)<\/mo><\/mrow><\/mfrac><\/mrow><annotation encoding=\"application\/x-tex\">a = \\frac{-(-1) \\pm \\sqrt{5}}{(2)(1)}<\/annotation><\/semantics><\/math><br><br><math data-latex=\"a = \\frac{1 \\pm \\sqrt{5}}{2}\"><semantics><mrow><mi>a<\/mi><mo>=<\/mo><mfrac><mrow><mn>1<\/mn><mo>\u00b1<\/mo><msqrt><mn>5<\/mn><\/msqrt><\/mrow><mn>2<\/mn><\/mfrac><\/mrow><annotation encoding=\"application\/x-tex\">a = \\frac{1 \\pm \\sqrt{5}}{2}<\/annotation><\/semantics><\/math><br><br>So the solutions are:<br><br><math data-latex=\"a = \\frac{1 + \\sqrt{5}}{2},\\ \\ \\ a = \\frac{1 - \\sqrt{5}}{2}\"><semantics><mrow><mi>a<\/mi><mo>=<\/mo><mfrac><mrow><mn>1<\/mn><mo>+<\/mo><msqrt><mn>5<\/mn><\/msqrt><\/mrow><mn>2<\/mn><\/mfrac><mo separator=\"true\">,<\/mo><mtext>&nbsp;<\/mtext><mtext>&nbsp;<\/mtext><mtext>&nbsp;<\/mtext><mi>a<\/mi><mo>=<\/mo><mfrac><mrow><mn>1<\/mn><mo>\u2212<\/mo><msqrt><mn>5<\/mn><\/msqrt><\/mrow><mn>2<\/mn><\/mfrac><\/mrow><annotation encoding=\"application\/x-tex\">a = \\frac{1 + \\sqrt{5}}{2},\\ \\ \\ a = \\frac{1 &#8211; \\sqrt{5}}{2}<\/annotation><\/semantics><\/math><br><br>\u2705 Correct Answer: <strong>C<\/strong><\/p>\n\n\n\n<p class=\"is-style-error\" style=\"font-size:0.9em\"><strong>Option A \u274c<\/strong><br>Trap: Student guesses integer roots without checking.<\/p>\n\n\n\n<p class=\"is-style-error\" style=\"font-size:0.9em\"><strong>Option B \u274c<\/strong><br>Trap: Student forces rational roots that don\u2019t satisfy the equation.<\/p>\n\n\n\n<p class=\"is-style-error\" style=\"font-size:0.9em\"><strong>Option D \u274c<\/strong><br>Trap: Student forgets that <math><semantics><mrow><mo>\u2212<\/mo><mi>b<\/mi><mo>=<\/mo><mo>+<\/mo><mn>1<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">-b = +1<\/annotation><\/semantics><\/math>, not \u22121.<\/p>\n\n\n\n<p class=\"is-style-info\" style=\"font-size:0.9em\"><strong>Desmos Steps<br><\/strong>1. Desmos don&#8217;t get <math data-latex=\"a\"><semantics><mi>a<\/mi><annotation encoding=\"application\/x-tex\">a<\/annotation><\/semantics><\/math> , so it will show nothing: <s>a^2 &#8211; a &#8211; 1 = 0<\/s><br>2. But it gets x and y<br>3. Enter: x^2 &#8211; x &#8211; 1<br>4. Desmos will graph this equation on the coordinate plane (<math data-latex=\"a\"><semantics><mi>a<\/mi><annotation encoding=\"application\/x-tex\">a<\/annotation><\/semantics><\/math> on x-axis, y = 0 line).<br>5. Graph shows:<br>~ In Desmos, <strong>click on the intersection points<\/strong> where the graph crosses the x-axis.<br>~ (1.61803, 0) = (x, y)<br>~ (-0.61803, 0) = (x, y)<br>~ we know, in Desmos, we replace <strong>a to x<\/strong><br>6. Now solve all options one-by-one to match the intersection points<br>7. Write options exactly like this &#8211; for example Option C<br>~ 1+ sqrt5 = it gives this 3.2360679775<br>~ copy it and paste it separate line then type: \/2<br>~ 3.2360679775\/2<br>~ You will get 1.61803&#8230;<br>~ That means it is correct, same step 7 &#8211; do it for 1-sqrt5  <br><strong>The intersect points match to Option C, done.<\/strong><\/p>\n<\/div><\/details><\/div>\n\n\n\n<div class=\"wp-block-coblocks-accordion-item\"><details><summary class=\"wp-block-coblocks-accordion-item__title\"><strong>7th Question<\/strong><\/summary><div class=\"wp-block-coblocks-accordion-item__content\">\n<p class=\"is-style-warning\" style=\"font-size:0.9em\"><strong>Question:<\/strong> If <math data-latex=\"\\frac{x}{y} = 4\"><semantics><mrow><mfrac><mi>x<\/mi><mi>y<\/mi><\/mfrac><mo>=<\/mo><mn>4<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">\\frac{x}{y} = 4<\/annotation><\/semantics><\/math> and <math data-latex=\"\\frac{24x}{ny} = 4\"><semantics><mrow><mfrac><mrow><mn>24<\/mn><mi>x<\/mi><\/mrow><mrow><mi>n<\/mi><mi>y<\/mi><\/mrow><\/mfrac><mo>=<\/mo><mn>4<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">\\frac{24x}{ny} = 4<\/annotation><\/semantics><\/math>, what is the value of <math data-latex=\"n\"><semantics><mi>n<\/mi><annotation encoding=\"application\/x-tex\">n<\/annotation><\/semantics><\/math>?<br>A) 4<br>B) 6<br>C) 24<br>D) 96<\/p>\n\n\n\n<p class=\"is-style-info\" style=\"font-size:0.9em\"><strong>Important Algebra Rule<\/strong><br>If two fractions are equal:<math display=\"block\"><semantics><mrow><mfrac><mi>x<\/mi><mi>y<\/mi><\/mfrac><mo>=<\/mo><mfrac><mrow><mn>24<\/mn><mi>x<\/mi><\/mrow><mrow><mi>n<\/mi><mi>y<\/mi><\/mrow><\/mfrac><\/mrow><annotation encoding=\"application\/x-tex\">\\frac{x}{y} = \\frac{24x}{ny}<\/annotation><\/semantics><\/math><br>you can <strong>substitute or simplify using known ratios<\/strong>.<\/p>\n\n\n\n<p class=\"is-style-success\" style=\"font-size:0.9em\"><strong>Step-by-Step Solution<\/strong><br><strong>Step 1: Use the first equation<\/strong><math display=\"block\"><semantics><mrow><mfrac><mi>x<\/mi><mi>y<\/mi><\/mfrac><mo>=<\/mo><mn>4<\/mn><mo>\u21d2<\/mo><mi>x<\/mi><mo>=<\/mo><mn>4<\/mn><mi>y<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">\\frac{x}{y} = 4 \\Rightarrow x = 4y<\/annotation><\/semantics><\/math><br><strong>Step 2: Substitute into the second equation<\/strong><br><math display=\"block\"><semantics><mrow><mfrac><mrow><mn>24<\/mn><mi>x<\/mi><\/mrow><mrow><mi>n<\/mi><mi>y<\/mi><\/mrow><\/mfrac><mo>=<\/mo><mn>4<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">\\frac{24x}{ny} = 4<\/annotation><\/semantics><\/math><br>Replace <math><semantics><mrow><mi>x<\/mi><mo>=<\/mo><mn>4<\/mn><mi>y<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">x = 4y<\/annotation><\/semantics><\/math>:<math display=\"block\"><semantics><mrow><mfrac><mrow><mn>24<\/mn><mo stretchy=\"false\">(<\/mo><mn>4<\/mn><mi>y<\/mi><mo stretchy=\"false\">)<\/mo><\/mrow><mrow><mi>n<\/mi><mi>y<\/mi><\/mrow><\/mfrac><mo>=<\/mo><mn>4<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">\\frac{24(4y)}{ny} = 4<\/annotation><\/semantics><\/math><br><strong>Step 3: Simplify<\/strong><math display=\"block\"><semantics><mrow><mfrac><mrow><mn>96<\/mn><mi>y<\/mi><\/mrow><mrow><mi>n<\/mi><mi>y<\/mi><\/mrow><\/mfrac><mo>=<\/mo><mn>4<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">\\frac{96y}{ny} = 4<\/annotation><\/semantics><\/math><br>Cancel <math><semantics><mrow><mi>y<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">y<\/annotation><\/semantics><\/math>:<math display=\"block\"><semantics><mrow><mfrac><mn>96<\/mn><mi>n<\/mi><\/mfrac><mo>=<\/mo><mn>4<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">\\frac{96}{n} = 4<\/annotation><\/semantics><\/math><br><strong>Step 4: Solve for n<\/strong><br><math data-latex=\"\\\\ \\frac{96}{n} = 4 \\\\ \\\\ 96 = 4n \\\\ \\\\ n = \\frac{96}{4} \\\\ \\\\ n = 24\"><semantics><mtable columnalign=\"left\" rowspacing=\"0em\"><mtr><mtd style=\"text-align:left\"><mo><\/mo><\/mtd><\/mtr><mtr><mtd style=\"text-align:left\"><mrow><mfrac><mn>96<\/mn><mi>n<\/mi><\/mfrac><mo>=<\/mo><mn>4<\/mn><\/mrow><mo><\/mo><\/mtd><\/mtr><mtr><mtd style=\"text-align:left\"><mo><\/mo><\/mtd><\/mtr><mtr><mtd style=\"text-align:left\"><mrow><mn>96<\/mn><mo>=<\/mo><mn>4<\/mn><mi>n<\/mi><\/mrow><mo><\/mo><\/mtd><\/mtr><mtr><mtd style=\"text-align:left\"><mo><\/mo><\/mtd><\/mtr><mtr><mtd style=\"text-align:left\"><mrow><mi>n<\/mi><mo>=<\/mo><mfrac><mn>96<\/mn><mn>4<\/mn><\/mfrac><\/mrow><mo><\/mo><\/mtd><\/mtr><mtr><mtd style=\"text-align:left\"><mo><\/mo><\/mtd><\/mtr><mtr><mtd style=\"text-align:left\"><mrow><mi>n<\/mi><mo>=<\/mo><mn>24<\/mn><\/mrow><\/mtd><\/mtr><\/mtable><annotation encoding=\"application\/x-tex\">\\\\ \\frac{96}{n} = 4 \\\\ \\\\ 96 = 4n \\\\ \\\\ n = \\frac{96}{4} \\\\ \\\\ n = 24<\/annotation><\/semantics><\/math><br><br><strong>Correct Answer<\/strong><math display=\"block\"><semantics><mrow><menclose notation=\"box\"><mstyle scriptlevel=\"0\" displaystyle=\"false\"><mstyle scriptlevel=\"0\" displaystyle=\"false\"><mstyle scriptlevel=\"0\" displaystyle=\"true\"><mn>24<\/mn><\/mstyle><\/mstyle><\/mstyle><\/menclose><\/mrow><annotation encoding=\"application\/x-tex\">\\boxed{24}<\/annotation><\/semantics><\/math><br>\u2705 <strong>Option: 24<\/strong><\/p>\n\n\n\n<p class=\"is-style-error\" style=\"font-size:0.9em\"><strong>Why Other Options Are Incorrect<\/strong><br><strong>4<\/strong> \u274c Too small, doesn\u2019t balance equation<br><strong>6<\/strong> \u274c Arithmetic mismatch<br><strong>96<\/strong> \u274c Forgot to divide by 4<br><br><strong>Common Student Mistakes<\/strong><br>\u274c Cancelling incorrectly<br>\u274c Forgetting substitution<br>\u274c Solving too early<\/p>\n\n\n\n<p class=\"is-style-info\" style=\"font-size:0.9em\"><strong>Desmos Verification<\/strong><br>1. Type: x\/y = 4<br>2. Type in 2nd line: 24x\/ny = 4<br>3. Desmos will create a graph line.<br>4. Desmos asks to add a slider or inform the value of <strong>n<\/strong><br>5. Enter: n value from options one-by-one<br>6. When you enter n = 24, notice that the question equation graph line and Option C graph line, matches exactly. It overlaps the question graph line.<br>That is your answer.<\/p>\n<\/div><\/details><\/div>\n\n\n\n<div class=\"wp-block-coblocks-accordion-item\"><details><summary class=\"wp-block-coblocks-accordion-item__title\"><strong>8th Question<\/strong><\/summary><div class=\"wp-block-coblocks-accordion-item__content\">\n<figure class=\"wp-block-table\"><table class=\"has-fixed-layout\"><thead><tr><th class=\"has-text-align-center\" data-align=\"center\"><math data-latex=\"x\"><semantics><mi>x<\/mi><annotation encoding=\"application\/x-tex\">x<\/annotation><\/semantics><\/math><\/th><th class=\"has-text-align-center\" data-align=\"center\"><math data-latex=\"f(x)\"><semantics><mrow><mi>f<\/mi><mo form=\"prefix\" stretchy=\"false\">(<\/mo><mi>x<\/mi><mo form=\"postfix\" stretchy=\"false\">)<\/mo><\/mrow><annotation encoding=\"application\/x-tex\">f(x)<\/annotation><\/semantics><\/math><\/th><\/tr><\/thead><tbody><tr><td class=\"has-text-align-center\" data-align=\"center\">0<\/td><td class=\"has-text-align-center\" data-align=\"center\">-2<\/td><\/tr><tr><td class=\"has-text-align-center\" data-align=\"center\">2<\/td><td class=\"has-text-align-center\" data-align=\"center\">4<\/td><\/tr><tr><td class=\"has-text-align-center\" data-align=\"center\">6<\/td><td class=\"has-text-align-center\" data-align=\"center\">16<\/td><\/tr><\/tbody><\/table><\/figure>\n\n\n\n<p class=\"is-style-warning\" style=\"font-size:0.9em\"><strong>Question:<\/strong> Some values of the linear function <math data-latex=\"f\"><semantics><mi>f<\/mi><annotation encoding=\"application\/x-tex\">f<\/annotation><\/semantics><\/math> are shown in the table above. What is the value of <math data-latex=\"f(3)\"><semantics><mrow><mi>f<\/mi><mo form=\"prefix\" stretchy=\"false\">(<\/mo><mn>3<\/mn><mo form=\"postfix\" stretchy=\"false\">)<\/mo><\/mrow><annotation encoding=\"application\/x-tex\">f(3)<\/annotation><\/semantics><\/math>?<br>A) 6<br>B) 7<br>C) 8<br>D) 9<\/p>\n\n\n\n<p class=\"is-style-info\" style=\"font-size:0.9em\"><strong>\ud83e\udde0 Core Concept (Why slope matters)<br><\/strong>The question explicitly says <strong>linear function<\/strong>.<br>That tells us <strong>one critical fact<\/strong>:<br>~ The rate of change (slope) is constant everywhere.<br>So if we find the slope once, we can find <strong>any missing value<\/strong>, including <math><semantics><mrow><mi>f<\/mi><mo stretchy=\"false\">(<\/mo><mn>3<\/mn><mo stretchy=\"false\">)<\/mo><\/mrow><annotation encoding=\"application\/x-tex\">f(3)<\/annotation><\/semantics><\/math>.<br><br>This table is <strong>not<\/strong> about random values.<br>Each pair <math><semantics><mrow><mo stretchy=\"false\">(<\/mo><mi>x<\/mi><mo separator=\"true\">,<\/mo><mi>f<\/mi><mo stretchy=\"false\">(<\/mo><mi>x<\/mi><mo stretchy=\"false\">)<\/mo><mo stretchy=\"false\">)<\/mo><\/mrow><annotation encoding=\"application\/x-tex\">(x, f(x))<\/annotation><\/semantics><\/math> represents a <strong>point on the graph<\/strong> of the function:<br>~ When <math><semantics><mrow><mi>x<\/mi><mo>=<\/mo><mn>0<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">x = 0<\/annotation><\/semantics><\/math>, <math><semantics><mrow><mi>f<\/mi><mo stretchy=\"false\">(<\/mo><mi>x<\/mi><mo stretchy=\"false\">)<\/mo><mo>=<\/mo><mo>\u2212<\/mo><mn>2<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">f(x) = -2<\/annotation><\/semantics><\/math> \u2192 point <strong>(0, \u22122)<\/strong><br>~ When <math><semantics><mrow><mi>x<\/mi><mo>=<\/mo><mn>2<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">x = 2<\/annotation><\/semantics><\/math>, <math><semantics><mrow><mi>f<\/mi><mo stretchy=\"false\">(<\/mo><mi>x<\/mi><mo stretchy=\"false\">)<\/mo><mo>=<\/mo><mn>4<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">f(x) = 4<\/annotation><\/semantics><\/math> \u2192 point <strong>(2, 4)<\/strong><br>~ When <math><semantics><mrow><mi>x<\/mi><mo>=<\/mo><mn>6<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">x = 6<\/annotation><\/semantics><\/math>, <math><semantics><mrow><mi>f<\/mi><mo stretchy=\"false\">(<\/mo><mi>x<\/mi><mo stretchy=\"false\">)<\/mo><mo>=<\/mo><mn>16<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">f(x) = 16<\/annotation><\/semantics><\/math> \u2192 point <strong>(6, 16)<\/strong><br>So we are <strong>not inventing<\/strong> anything like \u201c<math><semantics><mrow><msub><mi>y<\/mi><mn>1<\/mn><\/msub><\/mrow><annotation encoding=\"application\/x-tex\">y_1<\/annotation><\/semantics><\/math>\u200b, <math><semantics><mrow><msub><mi>y<\/mi><mn>2<\/mn><\/msub><\/mrow><annotation encoding=\"application\/x-tex\">y_2<\/annotation><\/semantics><\/math>\u200b\u201d.<br>~ Since it is Math, we can assume <math data-latex=\"f(x)\"><semantics><mrow><mi>f<\/mi><mo form=\"prefix\" stretchy=\"false\">(<\/mo><mi>x<\/mi><mo form=\"postfix\" stretchy=\"false\">)<\/mo><\/mrow><annotation encoding=\"application\/x-tex\">f(x)<\/annotation><\/semantics><\/math> point as <math data-latex=\"y\"><semantics><mi>y<\/mi><annotation encoding=\"application\/x-tex\">y<\/annotation><\/semantics><\/math>.<br>~ Like <math data-latex=\"(0, -2) = (x, y)\"><semantics><mrow><mo form=\"prefix\" stretchy=\"false\">(<\/mo><mn>0<\/mn><mo separator=\"true\">,<\/mo><mo form=\"prefix\" stretchy=\"false\">\u2212<\/mo><mn>2<\/mn><mo form=\"postfix\" stretchy=\"false\">)<\/mo><mo>=<\/mo><mo form=\"prefix\" stretchy=\"false\">(<\/mo><mi>x<\/mi><mo separator=\"true\">,<\/mo><mi>y<\/mi><mo form=\"postfix\" stretchy=\"false\">)<\/mo><\/mrow><annotation encoding=\"application\/x-tex\">(0, -2) = (x, y)<\/annotation><\/semantics><\/math><br>We are simply using the <strong>definition of a function<\/strong>:<br>~ A function assigns <strong>one output<\/strong> <math><semantics><mrow><mi>f<\/mi><mo stretchy=\"false\">(<\/mo><mi>x<\/mi><mo stretchy=\"false\">)<\/mo><\/mrow><annotation encoding=\"application\/x-tex\">f(x)<\/annotation><\/semantics><\/math> to <strong>each input<\/strong> <math><semantics><mrow><mi>x<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">x<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<p class=\"is-style-success\" style=\"font-size:0.9em\"><strong>\ud83e\uddee Step-by-Step Solution<br>Step 1: Find the slope using two points<br><\/strong>Using <math><semantics><mrow><mo stretchy=\"false\">(<\/mo><mn>0<\/mn><mo separator=\"true\">,<\/mo><mo>\u2212<\/mo><mn>2<\/mn><mo stretchy=\"false\">)<\/mo><\/mrow><annotation encoding=\"application\/x-tex\">(0, -2)<\/annotation><\/semantics><\/math> and <math><semantics><mrow><mo stretchy=\"false\">(<\/mo><mn>2<\/mn><mo separator=\"true\">,<\/mo><mn>4<\/mn><mo stretchy=\"false\">)<\/mo><\/mrow><annotation encoding=\"application\/x-tex\">(2, 4)<\/annotation><\/semantics><\/math>:<br>~ <math data-latex=\"x: 0, 2.\"><semantics><mrow><mi>x<\/mi><mo lspace=\"0.2222em\" rspace=\"0.2222em\">:<\/mo><mn>0<\/mn><mo separator=\"true\">,<\/mo><mn>2.<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">x: 0, 2.<\/annotation><\/semantics><\/math><br>~ <math data-latex=\"f(x)\\ or\\ y = -2, 4. \"><semantics><mrow><mi>f<\/mi><mo form=\"prefix\" stretchy=\"false\">(<\/mo><mi>x<\/mi><mo form=\"postfix\" stretchy=\"false\">)<\/mo><mtext>&nbsp;<\/mtext><mi>o<\/mi><mi>r<\/mi><mtext>&nbsp;<\/mtext><mi>y<\/mi><mo>=<\/mo><mo form=\"prefix\" stretchy=\"false\">\u2212<\/mo><mn>2<\/mn><mo separator=\"true\">,<\/mo><mn>4.<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">f(x)\\ or\\ y = -2, 4. <\/annotation><\/semantics><\/math><br>Slope formula:<br><math data-latex=\"m = \\frac{y_2 - x_2}{y_1 - x_1}\"><semantics><mrow><mi>m<\/mi><mo>=<\/mo><mfrac><mrow><msub><mi>y<\/mi><mn>2<\/mn><\/msub><mo>\u2212<\/mo><msub><mi>x<\/mi><mn>2<\/mn><\/msub><\/mrow><mrow><msub><mi>y<\/mi><mn>1<\/mn><\/msub><mo>\u2212<\/mo><msub><mi>x<\/mi><mn>1<\/mn><\/msub><\/mrow><\/mfrac><\/mrow><annotation encoding=\"application\/x-tex\">m = \\frac{y_2 &#8211; x_2}{y_1 &#8211; x_1}<\/annotation><\/semantics><\/math>  or  <math data-latex=\"m = \\frac{change\\ in\\ f(x)}{change\\ in\\ x}\"><semantics><mrow><mi>m<\/mi><mo>=<\/mo><mfrac><mrow><mi>c<\/mi><mi>h<\/mi><mi>a<\/mi><mi>n<\/mi><mi>g<\/mi><mi>e<\/mi><mtext>&nbsp;<\/mtext><mi>i<\/mi><mi>n<\/mi><mtext>&nbsp;<\/mtext><mi>f<\/mi><mo form=\"prefix\" stretchy=\"false\">(<\/mo><mi>x<\/mi><mo form=\"postfix\" stretchy=\"false\" lspace=\"0em\" rspace=\"0em\">)<\/mo><\/mrow><mrow><mi>c<\/mi><mi>h<\/mi><mi>a<\/mi><mi>n<\/mi><mi>g<\/mi><mi>e<\/mi><mtext>&nbsp;<\/mtext><mi>i<\/mi><mi>n<\/mi><mtext>&nbsp;<\/mtext><mi>x<\/mi><\/mrow><\/mfrac><\/mrow><annotation encoding=\"application\/x-tex\">m = \\frac{change\\ in\\ f(x)}{change\\ in\\ x}<\/annotation><\/semantics><\/math><br><br>Use <strong>any two points<\/strong> from the table (because the function is linear).<br><br><math display=\"block\"><semantics><mrow><mi>m<\/mi><mo>=<\/mo><mfrac><mrow><mn>4<\/mn><mo>\u2212<\/mo><mo stretchy=\"false\">(<\/mo><mo>\u2212<\/mo><mn>2<\/mn><mo stretchy=\"false\">)<\/mo><\/mrow><mrow><mn>2<\/mn><mo>\u2212<\/mo><mn>0<\/mn><\/mrow><\/mfrac><mo>=<\/mo><mfrac><mn>6<\/mn><mn>2<\/mn><\/mfrac><mo>=<\/mo><mn>3<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">m = \\frac{4 &#8211; (-2)}{2 &#8211; 0} = \\frac{6}{2} = 3<\/annotation><\/semantics><\/math><br>So the function increases by <strong>3 units in output<\/strong> for every <strong>1 unit increase in x<\/strong>.<br><br><strong>Step 3: What the Slope Means Here (Concept Check)<\/strong><br>Slope <math><semantics><mrow><mi>m<\/mi><mo>=<\/mo><mn>3<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">m = 3<\/annotation><\/semantics><\/math> means:<br>~ Increase <math><semantics><mrow><mi>x<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">x<\/annotation><\/semantics><\/math> by 1 \u2192 <math><semantics><mrow><mi>f<\/mi><mo stretchy=\"false\">(<\/mo><mi>x<\/mi><mo stretchy=\"false\">)<\/mo><\/mrow><annotation encoding=\"application\/x-tex\">f(x)<\/annotation><\/semantics><\/math> increases by 3<br>~ Increase <math><semantics><mrow><mi>x<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">x<\/annotation><\/semantics><\/math> by 2 \u2192 <math><semantics><mrow><mi>f<\/mi><mo stretchy=\"false\">(<\/mo><mi>x<\/mi><mo stretchy=\"false\">)<\/mo><\/mrow><annotation encoding=\"application\/x-tex\">f(x)<\/annotation><\/semantics><\/math> increases by 6<br>This interpretation is <strong>essential<\/strong> for the next step.<br><br><strong>Step 4: Move From a Known Value to <\/strong><math><semantics><mrow><mi>x<\/mi><mo>=<\/mo><mn>3<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">x = 3<\/annotation><\/semantics><\/math><br>We already know:<br><math display=\"block\"><semantics><mrow><mi>f<\/mi><mo stretchy=\"false\">(<\/mo><mn>2<\/mn><mo stretchy=\"false\">)<\/mo><mo>=<\/mo><mn>4<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">f(2) = 4<\/annotation><\/semantics><\/math><br>We want:<br><math display=\"block\"><semantics><mrow><mi>f<\/mi><mo stretchy=\"false\">(<\/mo><mn>3<\/mn><mo stretchy=\"false\">)<\/mo><\/mrow><annotation encoding=\"application\/x-tex\">f(3)<\/annotation><\/semantics><\/math><br>That is an increase of <strong>1<\/strong> in <math><semantics><mrow><mi>x<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">x<\/annotation><\/semantics><\/math>.<br>Since slope = 3:<br><math display=\"block\"><semantics><mrow><mi>f<\/mi><mo stretchy=\"false\">(<\/mo><mn>3<\/mn><mo stretchy=\"false\">)<\/mo><mo>=<\/mo><mn>4<\/mn><mo>+<\/mo><mn>3<\/mn><mo>=<\/mo><mn>7<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">f(3) = 4 + 3 = 7<\/annotation><\/semantics><\/math><br>\u2705 Correct Answer: <strong>B) 7<\/strong><\/p>\n\n\n\n<p class=\"is-style-error\" style=\"font-size:0.9em\"><strong>Option A: 6 \u274c<\/strong><br><strong>Trap:<\/strong> Student assumes slope = 2 instead of 3<br>(They divide incorrectly or eyeball the table)<\/p>\n\n\n\n<p class=\"is-style-error\" style=\"font-size:0.9em\"><strong>Option C: 8 \u274c<\/strong><br><strong>Trap:<\/strong> Student adds 4 (confuses output difference with input jump)<\/p>\n\n\n\n<p class=\"is-style-error\" style=\"font-size:0.9em\"><strong>Option D: 9 \u274c<\/strong><br><strong>Trap:<\/strong> Student multiplies slope by <math><semantics><mrow><mi>x<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">x<\/annotation><\/semantics><\/math> instead of applying change<\/p>\n\n\n\n<p class=\"is-style-info\" style=\"font-size:0.9em\"><strong>\ud83e\uddee Desmos Confirmation (Not the Reasoning)<\/strong><br>1. Open Desmos<br>2. Enter a <strong>Table<\/strong> (This is critical)<br>~ In Desmos, click <strong>\u201c+\u201d \u2192 Table<\/strong>, then enter: [Don&#8217;t try to change y into f(x)]<br>x    y<br>0   -2<br>2    4<br>6    16<br>\ud83d\udc49 Desmos will automatically plot the points.<br>3. Use <strong>Linear Regression (Best SAT Trick)<\/strong><br>~ In a new expression line, type <strong>exactly<\/strong>: y1 ~ mx1 + b<br>Desmos will output values for:<br><math><semantics><mrow><mi>m<\/mi><mo>=<\/mo><mn>3<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">m = 3<\/annotation><\/semantics><\/math><br><math><semantics><mrow><mi>b<\/mi><mo>=<\/mo><mo>\u2212<\/mo><mn>2<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">b = -2<\/annotation><\/semantics><\/math><br>So Desmos confirms:<math display=\"block\"><semantics><mrow><mi>f<\/mi><mo stretchy=\"false\">(<\/mo><mi>x<\/mi><mo stretchy=\"false\">)<\/mo><mo>=<\/mo><mn>3<\/mn><mi>x<\/mi><mo>\u2212<\/mo><mn>2<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">f(x) = 3x &#8211; 2<\/annotation><\/semantics><\/math><br>4. Evaluate <math><semantics><mrow><mi>f<\/mi><mo stretchy=\"false\">(<\/mo><mn>3<\/mn><mo stretchy=\"false\">)<\/mo><\/mrow><annotation encoding=\"application\/x-tex\">f(3)<\/annotation><\/semantics><\/math> Directly in Desmos: f(3) = 3(3) &#8211; 2<br><strong>Desmon outputs: 7<\/strong><\/p>\n<\/div><\/details><\/div>\n\n\n\n<div class=\"wp-block-coblocks-accordion-item\"><details><summary class=\"wp-block-coblocks-accordion-item__title\"><strong>9th Question<\/strong><\/summary><div class=\"wp-block-coblocks-accordion-item__content\">\n<figure class=\"wp-block-image size-full is-resized\"><img decoding=\"async\" src=\"https:\/\/us.mrenglishkj.com\/sat\/sat\/wp-content\/uploads\/2026\/01\/image_2026-01-18_085655053.png\" alt=\"Solve all math problems of SAT and learn from masters. Simple tricks, all tests for free\" class=\"wp-image-8934\" style=\"aspect-ratio:1.0148982374627544;width:336px;height:auto\"\/><\/figure>\n\n\n\n<p class=\"is-style-warning\" style=\"font-size:0.9em\"><strong>Question:<\/strong> In the <math data-latex=\"xy\"><semantics><mrow><mi>x<\/mi><mi>y<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">xy<\/annotation><\/semantics><\/math>-plane shown, square <math data-latex=\"ABCD\"><semantics><mrow><mi>A<\/mi><mi>B<\/mi><mi>C<\/mi><mi>D<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">ABCD<\/annotation><\/semantics><\/math> has its diagonals on the <math data-latex=\"x\"><semantics><mi>x<\/mi><annotation encoding=\"application\/x-tex\">x<\/annotation><\/semantics><\/math>&#8211; and <math data-latex=\"y\"><semantics><mi>y<\/mi><annotation encoding=\"application\/x-tex\">y<\/annotation><\/semantics><\/math>-axes. What is the area, in square units, of the square?<br>A) 20<br>B) 25<br>C) 50<br>D) 100<\/p>\n\n\n\n<p class=\"is-style-info\" style=\"font-size:0.9em\"><strong>Understanding the Diagram<\/strong><br>Square <strong>ABCD<\/strong><br><strong>Diagonals lie on x- and y-axes<\/strong><br>The square is centered at the origin<br>Coordinates shown:<br>~ Top point at (0, 5)<br>~ Right point at (5, 0)<br>~ Bottom point at (0, \u20135)<br>~ Left point at (\u20135, 0)<br><br><strong>Key Geometry Rule<\/strong><br>For a square:<br>Diagonal length <math><semantics><mrow><mi>d<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">d<\/annotation><\/semantics><\/math><br>Side length:<math display=\"block\"><semantics><mrow><mi>s<\/mi><mo>=<\/mo><mfrac><mi>d<\/mi><msqrt><mn>2<\/mn><\/msqrt><\/mfrac><\/mrow><annotation encoding=\"application\/x-tex\">s = \\frac{d}{\\sqrt{2}}<\/annotation><\/semantics><\/math><br>Area:<math display=\"block\"><semantics><mrow><mi>A<\/mi><mo>=<\/mo><msup><mi>s<\/mi><mn>2<\/mn><\/msup><\/mrow><annotation encoding=\"application\/x-tex\">A = s^2<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<p class=\"is-style-success\" style=\"font-size:0.9em\"><strong>Step-by-Step Solution<\/strong><br><strong>Step 1: Length of Diagonal<\/strong><br>From (0,5) to (0,\u20135):<math display=\"block\"><semantics><mrow><mi>d<\/mi><mo>=<\/mo><mn>10<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">d = 10<\/annotation><\/semantics><\/math><br><strong>Step 2: Side Length<\/strong><math display=\"block\"><semantics><mrow><mi>s<\/mi><mo>=<\/mo><mfrac><mn>10<\/mn><msqrt><mn>2<\/mn><\/msqrt><\/mfrac><mo>=<\/mo><mn>5<\/mn><msqrt><mn>2<\/mn><\/msqrt><\/mrow><annotation encoding=\"application\/x-tex\">s = \\frac{10}{\\sqrt{2}} = 5\\sqrt{2}<\/annotation><\/semantics><\/math><br><strong>Step 3: Area<\/strong><math display=\"block\"><semantics><mrow><mi>A<\/mi><mo>=<\/mo><mo stretchy=\"false\">(<\/mo><mn>5<\/mn><msqrt><mn>2<\/mn><\/msqrt><msup><mo stretchy=\"false\">)<\/mo><mn>2<\/mn><\/msup><mo>=<\/mo><mn>25<\/mn><mo>\u00d7<\/mo><mn>2<\/mn><mo>=<\/mo><mn>50<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">A = (5\\sqrt{2})^2 = 25 \\times 2 = 50<\/annotation><\/semantics><\/math><br>OR<br><math data-latex=\"s = \\frac{10}{\\sqrt2} \\\\ \\\\ A = (\\frac{10}{\\sqrt2})^2 \\\\ \\\\ A = \\frac{100}{2} \\\\ \\\\ A = 50\"><semantics><mtable columnalign=\"left\" rowspacing=\"0em\"><mtr><mtd style=\"text-align:left\"><mrow><mi>s<\/mi><mo>=<\/mo><mfrac><mn>10<\/mn><msqrt><mn>2<\/mn><\/msqrt><\/mfrac><\/mrow><mo><\/mo><\/mtd><\/mtr><mtr><mtd style=\"text-align:left\"><mo><\/mo><\/mtd><\/mtr><mtr><mtd style=\"text-align:left\"><mrow><mi>A<\/mi><mo>=<\/mo><mo form=\"prefix\" stretchy=\"false\">(<\/mo><mfrac><mn>10<\/mn><msqrt><mn>2<\/mn><\/msqrt><\/mfrac><msup><mo form=\"postfix\" stretchy=\"false\">)<\/mo><mn>2<\/mn><\/msup><\/mrow><mo><\/mo><\/mtd><\/mtr><mtr><mtd style=\"text-align:left\"><mo><\/mo><\/mtd><\/mtr><mtr><mtd style=\"text-align:left\"><mrow><mi>A<\/mi><mo>=<\/mo><mfrac><mn>100<\/mn><mn>2<\/mn><\/mfrac><\/mrow><mo><\/mo><\/mtd><\/mtr><mtr><mtd style=\"text-align:left\"><mo><\/mo><\/mtd><\/mtr><mtr><mtd style=\"text-align:left\"><mrow><mi>A<\/mi><mo>=<\/mo><mn>50<\/mn><\/mrow><\/mtd><\/mtr><\/mtable><annotation encoding=\"application\/x-tex\">s = \\frac{10}{\\sqrt2} \\\\ \\\\ A = (\\frac{10}{\\sqrt2})^2 \\\\ \\\\ A = \\frac{100}{2} \\\\ \\\\ A = 50<\/annotation><\/semantics><\/math><br><br><strong>Correct Answer: Option C<\/strong><math display=\"block\"><semantics><mrow><menclose notation=\"box\"><mstyle scriptlevel=\"0\" displaystyle=\"false\"><mstyle scriptlevel=\"0\" displaystyle=\"false\"><mstyle scriptlevel=\"0\" displaystyle=\"true\"><mn>50<\/mn><\/mstyle><\/mstyle><\/mstyle><\/menclose><\/mrow><annotation encoding=\"application\/x-tex\">\\boxed{50}<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<p class=\"is-style-error\" style=\"font-size:0.9em\"><strong>Why Other Options Are Incorrect<\/strong><br><strong>20<\/strong> \u274c too small<br><strong>25<\/strong> \u274c forgot diagonal relationship<br><strong>100<\/strong> \u274c squared diagonal incorrectly<br><br><strong>Common Student Traps<\/strong><br>Forgetting the \u221a2 factor<br>Treating diagonal as side<\/p>\n\n\n\n<p class=\"is-style-info\" style=\"font-size:0.9em\">Use DESMOS as a calculator.<\/p>\n<\/div><\/details><\/div>\n\n\n\n<div class=\"wp-block-coblocks-accordion-item\"><details><summary class=\"wp-block-coblocks-accordion-item__title\"><strong>10th Question<\/strong><\/summary><div class=\"wp-block-coblocks-accordion-item__content\">\n<p class=\"is-style-warning\" style=\"font-size:0.9em\"><strong>Question:<\/strong> If <math data-latex=\"x^2 = a + b\"><semantics><mrow><msup><mi>x<\/mi><mn>2<\/mn><\/msup><mo>=<\/mo><mi>a<\/mi><mo>+<\/mo><mi>b<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">x^2 = a + b<\/annotation><\/semantics><\/math> and <math data-latex=\"y^2 = a + c\"><semantics><mrow><msup><mi>y<\/mi><mn>2<\/mn><\/msup><mo>=<\/mo><mi>a<\/mi><mo>+<\/mo><mi>c<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">y^2 = a + c<\/annotation><\/semantics><\/math>, which of the following is equal to <math data-latex=\"(x^2 - y^2)^2\"><semantics><mrow><mo form=\"prefix\" stretchy=\"false\">(<\/mo><msup><mi>x<\/mi><mn>2<\/mn><\/msup><mo>\u2212<\/mo><msup><mi>y<\/mi><mn>2<\/mn><\/msup><msup><mo form=\"postfix\" stretchy=\"false\">)<\/mo><mn>2<\/mn><\/msup><\/mrow><annotation encoding=\"application\/x-tex\">(x^2 &#8211; y^2)^2<\/annotation><\/semantics><\/math>?<br>A) <math data-latex=\"a^2 - 2ac + c^2\"><semantics><mrow><msup><mi>a<\/mi><mn>2<\/mn><\/msup><mo>\u2212<\/mo><mn>2<\/mn><mi>a<\/mi><mi>c<\/mi><mo>+<\/mo><msup><mi>c<\/mi><mn>2<\/mn><\/msup><\/mrow><annotation encoding=\"application\/x-tex\">a^2 &#8211; 2ac + c^2<\/annotation><\/semantics><\/math><br>B) <math data-latex=\"b^2 - 2bc + c^2\"><semantics><mrow><msup><mi>b<\/mi><mn>2<\/mn><\/msup><mo>\u2212<\/mo><mn>2<\/mn><mi>b<\/mi><mi>c<\/mi><mo>+<\/mo><msup><mi>c<\/mi><mn>2<\/mn><\/msup><\/mrow><annotation encoding=\"application\/x-tex\">b^2 &#8211; 2bc + c^2<\/annotation><\/semantics><\/math><br>C) <math data-latex=\"4a^2 - 4abc + c^2\"><semantics><mrow><mn>4<\/mn><msup><mi>a<\/mi><mn>2<\/mn><\/msup><mo>\u2212<\/mo><mn>4<\/mn><mi>a<\/mi><mi>b<\/mi><mi>c<\/mi><mo>+<\/mo><msup><mi>c<\/mi><mn>2<\/mn><\/msup><\/mrow><annotation encoding=\"application\/x-tex\">4a^2 &#8211; 4abc + c^2<\/annotation><\/semantics><\/math><br>D) <math data-latex=\"4a^2 - 2abc + b^2c^2\"><semantics><mrow><mn>4<\/mn><msup><mi>a<\/mi><mn>2<\/mn><\/msup><mo>\u2212<\/mo><mn>2<\/mn><mi>a<\/mi><mi>b<\/mi><mi>c<\/mi><mo>+<\/mo><msup><mi>b<\/mi><mn>2<\/mn><\/msup><msup><mi>c<\/mi><mn>2<\/mn><\/msup><\/mrow><annotation encoding=\"application\/x-tex\">4a^2 &#8211; 2abc + b^2c^2<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<p class=\"is-style-info\" style=\"font-size:0.9em\"><strong>Important Algebra Rules<\/strong><br>1. Difference of squares: <math data-latex=\"x^2 - y^2 = (a + b) - (a + c)\"><semantics><mrow><msup><mi>x<\/mi><mn>2<\/mn><\/msup><mo>\u2212<\/mo><msup><mi>y<\/mi><mn>2<\/mn><\/msup><mo>=<\/mo><mo form=\"prefix\" stretchy=\"false\">(<\/mo><mi>a<\/mi><mo>+<\/mo><mi>b<\/mi><mo form=\"postfix\" stretchy=\"false\">)<\/mo><mo>\u2212<\/mo><mo form=\"prefix\" stretchy=\"false\">(<\/mo><mi>a<\/mi><mo>+<\/mo><mi>c<\/mi><mo form=\"postfix\" stretchy=\"false\">)<\/mo><\/mrow><annotation encoding=\"application\/x-tex\">x^2 &#8211; y^2 = (a + b) &#8211; (a + c)<\/annotation><\/semantics><\/math><br><br>2. Square a binomial: <math data-latex=\"(u - v)^2 = u^2 - 2uv + v^2\"><semantics><mrow><mo form=\"prefix\" stretchy=\"false\">(<\/mo><mi>u<\/mi><mo>\u2212<\/mo><mi>v<\/mi><msup><mo form=\"postfix\" stretchy=\"false\">)<\/mo><mn>2<\/mn><\/msup><mo>=<\/mo><msup><mi>u<\/mi><mn>2<\/mn><\/msup><mo>\u2212<\/mo><mn>2<\/mn><mi>u<\/mi><mi>v<\/mi><mo>+<\/mo><msup><mi>v<\/mi><mn>2<\/mn><\/msup><\/mrow><annotation encoding=\"application\/x-tex\">(u &#8211; v)^2 = u^2 &#8211; 2uv + v^2<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<p class=\"is-style-success\" style=\"font-size:0.9em\"><strong>Step-by-Step Solution<\/strong><br><strong>Step 1: Substitute<\/strong><math display=\"block\"><semantics><mrow><msup><mi>x<\/mi><mn>2<\/mn><\/msup><mo>\u2212<\/mo><msup><mi>y<\/mi><mn>2<\/mn><\/msup><mo>=<\/mo><mo stretchy=\"false\">(<\/mo><mi>a<\/mi><mo>+<\/mo><mi>b<\/mi><mo stretchy=\"false\">)<\/mo><mo>\u2212<\/mo><mo stretchy=\"false\">(<\/mo><mi>a<\/mi><mo>+<\/mo><mi>c<\/mi><mo stretchy=\"false\">)<\/mo><\/mrow><annotation encoding=\"application\/x-tex\">x^2 &#8211; y^2 = (a+b) &#8211; (a+c)<\/annotation><\/semantics><\/math><br>Simplify: <math data-latex=\"(a + b) - (a + c)\\ \\to\\ a + b - a - c\"><semantics><mrow><mo form=\"prefix\" stretchy=\"false\">(<\/mo><mi>a<\/mi><mo>+<\/mo><mi>b<\/mi><mo form=\"postfix\" stretchy=\"false\">)<\/mo><mo>\u2212<\/mo><mo form=\"prefix\" stretchy=\"false\">(<\/mo><mi>a<\/mi><mo>+<\/mo><mi>c<\/mi><mo form=\"postfix\" stretchy=\"false\">)<\/mo><mtext>&nbsp;<\/mtext><mo>\u2192<\/mo><mtext>&nbsp;<\/mtext><mi>a<\/mi><mo>+<\/mo><mi>b<\/mi><mo>\u2212<\/mo><mi>a<\/mi><mo>\u2212<\/mo><mi>c<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">(a + b) &#8211; (a + c)\\ \\to\\ a + b &#8211; a &#8211; c<\/annotation><\/semantics><\/math><br><math display=\"block\"><semantics><mrow><msup><mi>x<\/mi><mn>2<\/mn><\/msup><mo>\u2212<\/mo><msup><mi>y<\/mi><mn>2<\/mn><\/msup><mo>=<\/mo><mi>b<\/mi><mo>\u2212<\/mo><mi>c<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">x^2 &#8211; y^2 = b &#8211; c<\/annotation><\/semantics><\/math><br><strong>Step 2: Square<\/strong>: We have found <math data-latex=\"x^2 - y^2\"><semantics><mrow><msup><mi>x<\/mi><mn>2<\/mn><\/msup><mo>\u2212<\/mo><msup><mi>y<\/mi><mn>2<\/mn><\/msup><\/mrow><annotation encoding=\"application\/x-tex\">x^2 &#8211; y^2<\/annotation><\/semantics><\/math> but the question asked <math data-latex=\"(x^2 - y^2)^2\"><semantics><mrow><mo form=\"prefix\" stretchy=\"false\">(<\/mo><msup><mi>x<\/mi><mn>2<\/mn><\/msup><mo>\u2212<\/mo><msup><mi>y<\/mi><mn>2<\/mn><\/msup><msup><mo form=\"postfix\" stretchy=\"false\">)<\/mo><mn>2<\/mn><\/msup><\/mrow><annotation encoding=\"application\/x-tex\">(x^2 &#8211; y^2)^2<\/annotation><\/semantics><\/math><br>So, we put a square on <strong>b &#8211; c<\/strong><math display=\"block\"><semantics><mrow><mo stretchy=\"false\">(<\/mo><mi>b<\/mi><mo>\u2212<\/mo><mi>c<\/mi><msup><mo stretchy=\"false\">)<\/mo><mn>2<\/mn><\/msup><\/mrow><annotation encoding=\"application\/x-tex\">(b &#8211; c)^2<\/annotation><\/semantics><\/math><br>Expand: <strong><em>Binominal formula<\/em><\/strong><math display=\"block\"><semantics><mrow><msup><mi>b<\/mi><mn>2<\/mn><\/msup><mo>\u2212<\/mo><mn>2<\/mn><mi>b<\/mi><mi>c<\/mi><mo>+<\/mo><msup><mi>c<\/mi><mn>2<\/mn><\/msup><\/mrow><annotation encoding=\"application\/x-tex\">b^2 &#8211; 2bc + c^2<\/annotation><\/semantics><\/math><br><strong>Correct Answer<\/strong><math display=\"block\"><semantics><mrow><menclose notation=\"box\"><mstyle scriptlevel=\"0\" displaystyle=\"false\"><mstyle scriptlevel=\"0\" displaystyle=\"false\"><mstyle scriptlevel=\"0\" displaystyle=\"true\"><mrow><msup><mi>b<\/mi><mn>2<\/mn><\/msup><mo>\u2212<\/mo><mn>2<\/mn><mi>b<\/mi><mi>c<\/mi><mo>+<\/mo><msup><mi>c<\/mi><mn>2<\/mn><\/msup><\/mrow><\/mstyle><\/mstyle><\/mstyle><\/menclose><\/mrow><annotation encoding=\"application\/x-tex\">\\boxed{b^2 &#8211; 2bc + c^2}<\/annotation><\/semantics><\/math><br>\u2705 <strong>Option B<\/strong><\/p>\n\n\n\n<p class=\"is-style-error\" style=\"font-size:0.9em\"><strong>Why Other Options Are Incorrect<\/strong><br><strong>Option A:<\/strong> uses <math><semantics><mrow><mi>a<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">a<\/annotation><\/semantics><\/math>, but <math><semantics><mrow><mi>a<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">a<\/annotation><\/semantics><\/math> cancels out<br><strong>Option C:<\/strong> introduces extra terms not present<br><strong>Option D:<\/strong> invalid algebraic structure<br><br><strong>Common Student Mistakes<\/strong><br>\u274c Expanding before simplifying<br>\u274c Forgetting cancellation of <math><semantics><mrow><mi>a<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">a<\/annotation><\/semantics><\/math><br>\u274c Squaring incorrectly<\/p>\n\n\n\n<p class=\"is-style-info\" style=\"font-size:0.9em\"><strong>DESMOS TRICKS<\/strong><br>1. Assign values to a, b, and c.<br>a = 1<br>b = 2<br>c = 3<br>~ You can assign any values.<br>2. Simplify: You must simplify it yourself to this point &#8211; take reference from above solution.<br>~ Type:<br>~~ No<br>(x^2 &#8211; y^2)^2 \u274c<br>((a + b) &#8211; (a + c))^2 \u274c<br>~~ Yes<br>((1 + 2) &#8211; (1 + 3))^2 \u2705<br>~ Output: 1<br>3. Type options values one-by-one:<br>~ Option B: 2^2 &#8211; 2*2*3 + 3^2<br>~ Output: 1<br>Option B and question equation matches the output.  <\/p>\n<\/div><\/details><\/div>\n\n\n\n<div class=\"wp-block-coblocks-accordion-item\"><details><summary class=\"wp-block-coblocks-accordion-item__title\"><strong>11th Question<\/strong><\/summary><div class=\"wp-block-coblocks-accordion-item__content\">\n<p class=\"is-style-warning\" style=\"font-size:0.9em\"><strong>Question:<\/strong> Line \ud835\udcc1 is defined by 3y + 12x = 5. Line <em>n<\/em> is perpendicular to line \ud835\udcc1 in the xy-plane. What is the slope of line <em>n<\/em>?<br><br>[Type-Based Answer: In the final exam, you will type the answer rather than choose from options.]<\/p>\n\n\n\n<p class=\"is-style-success\" style=\"font-size:0.9em\"><strong>1\/4 or 0.25<\/strong> is correct.<br><br><strong>\ud83e\uddee Step-by-Step Correct Solution<\/strong><br>The question asks for the <strong>slope of line <em>n<\/em><\/strong>, but line <math><semantics><mrow><mi>n<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">n<\/annotation><\/semantics><\/math> is not given directly. Instead, we are told an important relationship: <strong>line <em>n<\/em> is perpendicular to line \u2113<\/strong>.<br>That means the <strong>first necessary step<\/strong> is to find the <strong>slope of line \u2113<\/strong>.<br>Without knowing the slope of line <math><semantics><mrow><mi mathvariant=\"normal\">\u2113<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">\\ell<\/annotation><\/semantics><\/math>, we cannot determine the slope of a line perpendicular to it.<br><strong>Step 1: Rewrite the given equation in slope form<\/strong><br>Given:<br><math display=\"block\"><semantics><mrow><mn>3<\/mn><mi>y<\/mi><mo>+<\/mo><mn>12<\/mn><mi>x<\/mi><mo>=<\/mo><mn>5<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">3y + 12x = 5<\/annotation><\/semantics><\/math><br>To identify the slope, we need the equation in <strong>slope\u2013intercept form<\/strong>:<br><math display=\"block\"><semantics><mrow><mi>y<\/mi><mo>=<\/mo><mi>m<\/mi><mi>x<\/mi><mo>+<\/mo><mi>b<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">y = mx + b<\/annotation><\/semantics><\/math><br>This form is essential because:<br>~ The coefficient of <math><semantics><mrow><mi>x<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">x<\/annotation><\/semantics><\/math> (the number multiplying <math><semantics><mrow><mi>x<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">x<\/annotation><\/semantics><\/math>) is <strong>by definition the slope<\/strong><br>~ Slope is not directly visible unless the equation is written with <strong>y alone on one side<\/strong><br>We <strong>do not isolate x<\/strong> because:<br>~ The standard slope formula used in SAT is taken from <math><semantics><mrow><mi>y<\/mi><mo>=<\/mo><mi>m<\/mi><mi>x<\/mi><mo>+<\/mo><mi>b<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">y = mx + b<\/annotation><\/semantics><\/math><br>~ Isolating x would require extra conversion later and hides the slope<br>So we isolate <strong>y<\/strong> deliberately to expose the slope clearly.<br>Subtract <math><semantics><mrow><mn>12<\/mn><mi>x<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">12x<\/annotation><\/semantics><\/math> from both sides:<br><math display=\"block\"><semantics><mrow><mn>3<\/mn><mi>y<\/mi><mo>=<\/mo><mo>\u2212<\/mo><mn>12<\/mn><mi>x<\/mi><mo>+<\/mo><mn>5<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">3y = -12x + 5<\/annotation><\/semantics><\/math><br>Now divide <strong>every term<\/strong> by 3:<br><math display=\"block\"><semantics><mrow><mi>y<\/mi><mo>=<\/mo><mo>\u2212<\/mo><mn>4<\/mn><mi>x<\/mi><mo>+<\/mo><mstyle displaystyle=\"false\" scriptlevel=\"0\"><mfrac><mn>5<\/mn><mn>3<\/mn><\/mfrac><\/mstyle><\/mrow><annotation encoding=\"application\/x-tex\">y = -4x + \\tfrac{5}{3}<\/annotation><\/semantics><\/math><br><strong>Step 2: Identify the slope of line <\/strong><math><semantics><mrow><mi mathvariant=\"normal\">\u2113<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">\\ell<\/annotation><\/semantics><\/math><br>From:<br><math display=\"block\"><semantics><mrow><mi>y<\/mi><mo>=<\/mo><mo>\u2212<\/mo><mn>4<\/mn><mi>x<\/mi><mo>+<\/mo><mstyle displaystyle=\"false\" scriptlevel=\"0\"><mfrac><mn>5<\/mn><mn>3<\/mn><\/mfrac><\/mstyle><\/mrow><annotation encoding=\"application\/x-tex\">y = -4x + \\tfrac{5}{3}<\/annotation><\/semantics><\/math><br>The slope of line <math><semantics><mrow><mi mathvariant=\"normal\">\u2113<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">\\ell<\/annotation><\/semantics><\/math> is:<br><math display=\"block\"><semantics><mrow><msub><mi>m<\/mi><mi mathvariant=\"normal\">\u2113<\/mi><\/msub><mo>=<\/mo><mo>\u2212<\/mo><mn>4<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">m_{\\ell} = -4<\/annotation><\/semantics><\/math><br>This value is critical \u2014 everything that follows depends on it.<br><br><strong>Step 3: Why perpendicular lines use negative reciprocals<\/strong><br>A key SAT rule applies here: <strong>If two lines are perpendicular, the product of their slopes is \u22121<\/strong><br>Mathematically:<br><math display=\"block\"><semantics><mrow><msub><mi>m<\/mi><mi mathvariant=\"normal\">\u2113<\/mi><\/msub><mo>\u22c5<\/mo><msub><mi>m<\/mi><mi>n<\/mi><\/msub><mo>=<\/mo><mo>\u2212<\/mo><mn>1<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">m_{\\ell} \\cdot m_{n} = -1<\/annotation><\/semantics><\/math><br>This rule comes from coordinate geometry and is always used on SAT when:<br>~ The word <strong>perpendicular<\/strong> appears<br>~ Slopes are involved<br><br><strong>Step 4: Find the slope of line <\/strong><math><semantics><mrow><mi>n<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">n<\/annotation><\/semantics><\/math><br>We already know:<br><math display=\"block\"><semantics><mrow><msub><mi>m<\/mi><mi mathvariant=\"normal\">\u2113<\/mi><\/msub><mo>=<\/mo><mo>\u2212<\/mo><mn>4<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">m_{\\ell} = -4<\/annotation><\/semantics><\/math><br>Apply the perpendicular-slope rule:<br><math display=\"block\"><semantics><mrow><mo stretchy=\"false\">(<\/mo><mo>\u2212<\/mo><mn>4<\/mn><mo stretchy=\"false\">)<\/mo><mo stretchy=\"false\">(<\/mo><msub><mi>m<\/mi><mi>n<\/mi><\/msub><mo stretchy=\"false\">)<\/mo><mo>=<\/mo><mo>\u2212<\/mo><mn>1<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">(-4)(m_n) = -1<\/annotation><\/semantics><\/math><br>Solve for <math><semantics><mrow><msub><mi>m<\/mi><mi>n<\/mi><\/msub><\/mrow><annotation encoding=\"application\/x-tex\">m_n<\/annotation><\/semantics><\/math>\u200b:<br><math display=\"block\"><semantics><mrow><msub><mi>m<\/mi><mi>n<\/mi><\/msub><mo>=<\/mo><mtext>negative&nbsp;reciprocal&nbsp;of&nbsp;<\/mtext><mo>\u2212<\/mo><mn>4<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">m_n = \\text{negative reciprocal of } -4<\/annotation><\/semantics><\/math><br>That means:<br><math display=\"block\"><semantics><mrow><msub><mi>m<\/mi><mi>n<\/mi><\/msub><mo>=<\/mo><mstyle displaystyle=\"false\" scriptlevel=\"0\"><mfrac><mn>1<\/mn><mn>4<\/mn><\/mfrac><\/mstyle><\/mrow><annotation encoding=\"application\/x-tex\">m_n = \\tfrac{1}{4}<\/annotation><\/semantics><\/math><br>This process is often described as taking the <strong>negative reciprocal<\/strong>:<br><br>~ Reciprocal of <math><semantics><mrow><mo>\u2212<\/mo><mn>4<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">-4<\/annotation><\/semantics><\/math> \u2192 <math><semantics><mrow><mo>\u2212<\/mo><mfrac><mn>1<\/mn><mn>4<\/mn><\/mfrac><\/mrow><annotation encoding=\"application\/x-tex\">-\\tfrac{1}{4}<\/annotation><\/semantics><\/math><br><br>~ Change the sign \u2192 <math><semantics><mrow><mfrac><mn>1<\/mn><mn>4<\/mn><\/mfrac><\/mrow><annotation encoding=\"application\/x-tex\">\\tfrac{1}{4}<\/annotation><\/semantics><\/math><br><br>\u2714 This is the slope of line <math><semantics><mrow><mi>n<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">n<\/annotation><\/semantics><\/math><br>\u2714 Divide 1 by 4 and you will get 0.25<br>\u2714 So either 0.25 or 1\/4 is the correct answer<\/p>\n\n\n\n<p class=\"is-style-error\" style=\"font-size:0.9em\">\u274c Trap 1: Answer = <strong>\u22124<\/strong><br><strong>Why students choose this:<\/strong><br>~ They stop after finding the slope of line <math><semantics><mrow><mi mathvariant=\"normal\">\u2113<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">\\ell<\/annotation><\/semantics><\/math><br>~ They forget the question asks for the slope of <strong>line <em>n<\/em><\/strong><br>Mistake:<br>~ Ignoring the word <strong>perpendicular<\/strong><br>~ \u274c Trap 1: Answer = <strong>\u22124<\/strong><br><strong>Why students choose this:<\/strong><br>~ They stop after finding the slope of line <math><semantics><mrow><mi mathvariant=\"normal\">\u2113<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">\\ell<\/annotation><\/semantics><\/math><br>~ They forget the question asks for the slope of <strong>line <em>n<\/em><\/strong><br>Mistake:<br>~ Ignoring the word <strong>perpendicular<\/strong><br>~ Confusing \u201copposite slope\u201d with \u201cperpendicular slope\u201d<\/p>\n\n\n\n<p class=\"is-style-error\" style=\"font-size:0.9em\">\u274c Trap 2: Answer = <strong>\u22121\/4 or -0.25\u200b<\/strong><br><strong>Why students choose this:<\/strong><br>~ They correctly take the reciprocal<br>~ They forget to change the sign<br>Mistake: Incomplete application of the negative-reciprocal rule<\/p>\n\n\n\n<p class=\"is-style-info\" style=\"font-size:0.9em\"><strong>\ud83e\uddee DESMOS CALCULATOR \u2014 VERIFICATION METHOD<\/strong><br><strong>Step-by-Step in Desmos<\/strong><br>1. Open <strong>Desmos<\/strong><br>2. In <strong>Expression Line 1<\/strong>, type: y = -4x + 5\/3<br>(This is line <math><semantics><mrow><mi mathvariant=\"normal\">\u2113<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">\\ell<\/annotation><\/semantics><\/math>)<br>3. In <strong>Expression Line 2<\/strong>, type: y = (1\/4)x<br>(This represents a line with slope <math><semantics><mrow><mfrac><mn>1<\/mn><mn>4<\/mn><\/mfrac><\/mrow><annotation encoding=\"application\/x-tex\">\\tfrac{1}{4}<\/annotation><\/semantics><\/math>)<br>4. Zoom in near the intersection<br>5. Observe:<br>~ Lines intersect at a right angle<br>~ Confirms perpendicularity<br><br>\u2714 Confirms slope = <math><semantics><mrow><mfrac><mn>1<\/mn><mn>4<\/mn><\/mfrac><\/mrow><annotation encoding=\"application\/x-tex\">\\tfrac{1}{4}<\/annotation><\/semantics><\/math><\/p>\n<\/div><\/details><\/div>\n\n\n\n<div class=\"wp-block-coblocks-accordion-item\"><details><summary class=\"wp-block-coblocks-accordion-item__title\"><strong>12th Question<\/strong><\/summary><div class=\"wp-block-coblocks-accordion-item__content\">\n<p class=\"is-style-warning\" style=\"font-size:0.9em\"><strong>Question:<\/strong> Percent of Residents Who Earned a Bachelor&#8217;s Degree or Higher<\/p>\n\n\n\n<figure class=\"wp-block-table\"><table class=\"has-fixed-layout\"><thead><tr><th class=\"has-text-align-center\" data-align=\"center\">State<\/th><th class=\"has-text-align-center\" data-align=\"center\">Percent of residents<\/th><\/tr><\/thead><tbody><tr><td class=\"has-text-align-center\" data-align=\"center\">State A<\/td><td class=\"has-text-align-center\" data-align=\"center\">21.9%<\/td><\/tr><tr><td class=\"has-text-align-center\" data-align=\"center\">State B<\/td><td class=\"has-text-align-center\" data-align=\"center\">27.9%<\/td><\/tr><tr><td class=\"has-text-align-center\" data-align=\"center\">State C<\/td><td class=\"has-text-align-center\" data-align=\"center\">25.9%<\/td><\/tr><tr><td class=\"has-text-align-center\" data-align=\"center\">State D<\/td><td class=\"has-text-align-center\" data-align=\"center\">19.5%<\/td><\/tr><tr><td class=\"has-text-align-center\" data-align=\"center\">State E<\/td><td class=\"has-text-align-center\" data-align=\"center\">30.1%<\/td><\/tr><tr><td class=\"has-text-align-center\" data-align=\"center\">State F<\/td><td class=\"has-text-align-center\" data-align=\"center\">36.4%<\/td><\/tr><tr><td class=\"has-text-align-center\" data-align=\"center\">State G<\/td><td class=\"has-text-align-center\" data-align=\"center\">35.5%<\/td><\/tr><\/tbody><\/table><\/figure>\n\n\n\n<p class=\"is-style-warning\" style=\"font-size:0.9em\">A survey was given to residents of all 50 states asking if they had earned a bachelor\u2019s degree or higher. The results from 7 of the states are given in the table above. The median percent of residents who earned a bachelor\u2019s degree or higher for all 50 states was 26.95%. What is the difference between the median percent of residents who earned a bachelor\u2019s degree or higher for these 7 states and the median for all 50 states?<br>A) 0.05%<br>B) 0.95%<br>C) 1.22%<br>D) 7.45%<\/p>\n\n\n\n<p class=\"is-style-info\" style=\"font-size:0.9em\"><strong>Understanding the Question<\/strong><br>You are given <strong>percentages from 7 states<\/strong> and told that the <strong>median for all 50 states is 26.95%<\/strong>.<br>You must:<br>~ Find the <strong>median of the 7 given values<\/strong><br>~ Find the <strong>difference<\/strong> between the two medians<br><br><strong>Key Rule (Median)<\/strong><br>~ The <strong>median<\/strong> is the <strong>middle value<\/strong> when data is arranged in <strong>ascending order<\/strong><br>~ For <strong>7 numbers<\/strong>, the median is the <strong>4th number<\/strong><\/p>\n\n\n\n<p class=\"is-style-success\" style=\"font-size:0.9em\"><strong>Step-by-Step Solution<\/strong><br>Step 1: List the percentages in ascending order<br>19.5, 21.9, 25.9, <strong>27.9<\/strong>, 30.1, 35.5, 36.4<br><br>Step 2: Identify the median of the 7 states<br>4th value = <strong>27.9%<\/strong><br><br>Step 3: Compare with the national median<br>National median = <strong>26.95%<\/strong><br><br>Step 4: Find the difference<math display=\"block\"><semantics><mrow><mn>27.9<\/mn><mo>\u2212<\/mo><mn>26.95<\/mn><mo>=<\/mo><mn>0.95<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">27.9 &#8211; 26.95 = 0.95<\/annotation><\/semantics><\/math><br>\u2705 <strong>Correct Answer: 0.95%<\/strong><\/p>\n\n\n\n<p class=\"is-style-error\" style=\"font-size:0.9em\">\u274c Why the Other Options Are Wrong<br><strong>7.45%<\/strong> \u2192 comparing max and min, not medians<br><strong>0.05%<\/strong> \u2192 subtracting the wrong values<br><strong>1.22%<\/strong> \u2192 difference between wrong middle values<\/p>\n\n\n\n<p class=\"is-style-info\" style=\"font-size:0.9em\">Use DESMOS only a calculator.<\/p>\n<\/div><\/details><\/div>\n\n\n\n<div class=\"wp-block-coblocks-accordion-item\"><details><summary class=\"wp-block-coblocks-accordion-item__title\"><strong>13th Question<\/strong><\/summary><div class=\"wp-block-coblocks-accordion-item__content\">\n<p class=\"is-style-warning\" style=\"font-size:0.9em\"><strong>Question:<\/strong> An oceanographer uses the equation <math data-latex=\"s = \\frac{7}{4}p\"><semantics><mrow><mi>s<\/mi><mo>=<\/mo><mfrac><mn>7<\/mn><mn>4<\/mn><\/mfrac><mi>p<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">s = \\frac{7}{4}p<\/annotation><\/semantics><\/math> to model the speed <math data-latex=\"s\"><semantics><mi>s<\/mi><annotation encoding=\"application\/x-tex\">s<\/annotation><\/semantics><\/math>, in knots, of an ocean wave, where <math data-latex=\"p\"><semantics><mi>p<\/mi><annotation encoding=\"application\/x-tex\">p<\/annotation><\/semantics><\/math> represents the period of the wave, in seconds. Which of the following represents the period of the wave in terms of the speed of the wave?<br>A) <math data-latex=\"p = \\frac{4}{7}s\"><semantics><mrow><mi>p<\/mi><mo>=<\/mo><mfrac><mn>4<\/mn><mn>7<\/mn><\/mfrac><mi>s<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">p = \\frac{4}{7}s<\/annotation><\/semantics><\/math><br><br>B) <math data-latex=\"p = \\frac{7}{4}s\"><semantics><mrow><mi>p<\/mi><mo>=<\/mo><mfrac><mn>7<\/mn><mn>4<\/mn><\/mfrac><mi>s<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">p = \\frac{7}{4}s<\/annotation><\/semantics><\/math><br><br>C) <math data-latex=\"p = \\frac{4}{7} + s\"><semantics><mrow><mi>p<\/mi><mo>=<\/mo><mfrac><mn>4<\/mn><mn>7<\/mn><\/mfrac><mo>+<\/mo><mi>s<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">p = \\frac{4}{7} + s<\/annotation><\/semantics><\/math><br><br>D) <math data-latex=\"p = \\frac{7}{4} + s\"><semantics><mrow><mi>p<\/mi><mo>=<\/mo><mfrac><mn>7<\/mn><mn>4<\/mn><\/mfrac><mo>+<\/mo><mi>s<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">p = \\frac{7}{4} + s<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<p class=\"is-style-info\" style=\"font-size:0.9em\"><strong>Understand the Question<\/strong><br><strong>Problem (understanding the question)<\/strong><br>Given:<math display=\"block\"><semantics><mrow><mi>s<\/mi><mo>=<\/mo><mfrac><mn>7<\/mn><mn>4<\/mn><\/mfrac><mi>p<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">s = \\frac{7}{4}p<\/annotation><\/semantics><\/math><br>Where:<br><math><semantics><mrow><mi>s<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">s<\/annotation><\/semantics><\/math> = speed (knots)<br><math><semantics><mrow><mi>p<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">p<\/annotation><\/semantics><\/math> = period (seconds)<br>Asked:<br>\ud83d\udc49 <strong>Express p in terms of s<\/strong><br><br><strong>Important Rule<\/strong><br>To solve for a variable: Undo multiplication using <strong>division<\/strong><\/p>\n\n\n\n<p class=\"is-style-success\" style=\"font-size:0.9em\"><strong>Step-by-Step Solution<\/strong><math display=\"block\"><semantics><mrow><mi>s<\/mi><mo>=<\/mo><mfrac><mn>7<\/mn><mn>4<\/mn><\/mfrac><mi>p<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">s = \\frac{7}{4}p<\/annotation><\/semantics><\/math><br>Multiply both sides by <math><semantics><mrow><mfrac><mn>4<\/mn><mn>7<\/mn><\/mfrac><\/mrow><annotation encoding=\"application\/x-tex\">\\frac{4}{7}<\/annotation><\/semantics><\/math>\u200b:<math display=\"block\"><semantics><mrow><mfrac><mn>4<\/mn><mn>7<\/mn><\/mfrac><mi>s<\/mi><mo>=<\/mo><mi>p<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">\\frac{4}{7}s = p<\/annotation><\/semantics><\/math><br>Rewriting:<math display=\"block\"><semantics><mrow><mi>p<\/mi><mo>=<\/mo><mfrac><mn>4<\/mn><mn>7<\/mn><\/mfrac><mi>s<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">p = \\frac{4}{7}s<\/annotation><\/semantics><\/math><br>Or<br>There is another mathematical way:<br>We need to find <strong>p<\/strong><br><math data-latex=\"s = \\frac{7}{4}p \\\\ s = \\frac{7}{4} \\times p\"><semantics><mtable columnalign=\"left\" rowspacing=\"0em\"><mtr><mtd style=\"text-align:left\"><mrow><mi>s<\/mi><mo>=<\/mo><mfrac><mn>7<\/mn><mn>4<\/mn><\/mfrac><mi>p<\/mi><\/mrow><mo><\/mo><\/mtd><\/mtr><mtr><mtd style=\"text-align:left\"><mrow><mi>s<\/mi><mo>=<\/mo><mfrac><mn>7<\/mn><mn>4<\/mn><\/mfrac><mo>\u00d7<\/mo><mi>p<\/mi><\/mrow><\/mtd><\/mtr><\/mtable><annotation encoding=\"application\/x-tex\">s = \\frac{7}{4}p \\\\ s = \\frac{7}{4} \\times p<\/annotation><\/semantics><\/math><br>Numerator goes on divide of another side.<br>Denominator \/ value in divide goes another side as multiply<br>So:<br><math data-latex=\" \\\\ \\frac{4}{7}s = p \\\\ p = \\frac{4}{7}s\"><semantics><mtable columnalign=\"left\" rowspacing=\"0em\"><mtr><mtd style=\"text-align:left\"><mo><\/mo><\/mtd><\/mtr><mtr><mtd style=\"text-align:left\"><mrow><mfrac><mn>4<\/mn><mn>7<\/mn><\/mfrac><mi>s<\/mi><mo>=<\/mo><mi>p<\/mi><\/mrow><mo><\/mo><\/mtd><\/mtr><mtr><mtd style=\"text-align:left\"><mrow><mi>p<\/mi><mo>=<\/mo><mfrac><mn>4<\/mn><mn>7<\/mn><\/mfrac><mi>s<\/mi><\/mrow><\/mtd><\/mtr><\/mtable><annotation encoding=\"application\/x-tex\"> \\\\ \\frac{4}{7}s = p \\\\ p = \\frac{4}{7}s<\/annotation><\/semantics><\/math><br><br><strong>Correct Answer<\/strong><math display=\"block\"><semantics><mrow><menclose notation=\"box\"><mstyle scriptlevel=\"0\" displaystyle=\"false\"><mstyle scriptlevel=\"0\" displaystyle=\"false\"><mstyle scriptlevel=\"0\" displaystyle=\"true\"><mrow><mi>p<\/mi><mo>=<\/mo><mfrac><mn>4<\/mn><mn>7<\/mn><\/mfrac><mi>s<\/mi><\/mrow><\/mstyle><\/mstyle><\/mstyle><\/menclose><\/mrow><annotation encoding=\"application\/x-tex\">\\boxed{p = \\frac{4}{7}s}<\/annotation><\/semantics><\/math><br>\u2705 <strong>Option A<\/strong><\/p>\n\n\n\n<p class=\"is-style-error\" style=\"font-size:0.9em\"><strong>Why Other Options Are Incorrect<\/strong><br><strong>Option B:<\/strong> keeps the same multiplier \u2014 no rearranging<br><strong>Option C &amp; D:<\/strong> add instead of multiply \u2014 violates algebra rules<br><br><strong>Common Student Mistakes<\/strong><br>\u274c Switching variables without isolating<br>\u274c Adding instead of dividing<br>\u274c Forgetting inverse operations<\/p>\n<\/div><\/details><\/div>\n\n\n\n<div class=\"wp-block-coblocks-accordion-item\"><details><summary class=\"wp-block-coblocks-accordion-item__title\"><strong>14th Question<\/strong><\/summary><div class=\"wp-block-coblocks-accordion-item__content\">\n<figure class=\"wp-block-image size-full is-resized\"><img decoding=\"async\" src=\"https:\/\/us.mrenglishkj.com\/sat\/sat\/wp-content\/uploads\/2026\/01\/Screenshot-2026-01-13-232340.png\" alt=\"Simple explanations of trigonometry questions - learn math and take tests\" class=\"wp-image-8783\" style=\"aspect-ratio:1.7716128247247065;width:343px;height:auto\"\/><\/figure>\n\n\n\n<p class=\"is-style-warning\" style=\"font-size:0.9em\"><strong>Question:<\/strong> In the figure shown, points <math data-latex=\"Q, R, S\"><semantics><mrow><mi>Q<\/mi><mo separator=\"true\">,<\/mo><mi>R<\/mi><mo separator=\"true\">,<\/mo><mi>S<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">Q, R, S<\/annotation><\/semantics><\/math>, and <math data-latex=\"T\"><semantics><mi>T<\/mi><annotation encoding=\"application\/x-tex\">T<\/annotation><\/semantics><\/math> lie on line segment <math data-latex=\"PV\"><semantics><mrow><mi>P<\/mi><mi>V<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">PV<\/annotation><\/semantics><\/math>, and line segment <math data-latex=\"RU\"><semantics><mrow><mi>R<\/mi><mi>U<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">RU<\/annotation><\/semantics><\/math> intersects line segment <math data-latex=\"SX\"><semantics><mrow><mi>S<\/mi><mi>X<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">SX<\/annotation><\/semantics><\/math> at point <math data-latex=\"W\"><semantics><mi>W<\/mi><annotation encoding=\"application\/x-tex\">W<\/annotation><\/semantics><\/math>. The measure of <math data-latex=\"\\angle{SQX}\"><semantics><mrow><mi>\u2220<\/mi><mrow><mi>S<\/mi><mi>Q<\/mi><mi>X<\/mi><\/mrow><\/mrow><annotation encoding=\"application\/x-tex\">\\angle{SQX}<\/annotation><\/semantics><\/math> is <math data-latex=\"48^\\circ\"><semantics><msup><mn>48<\/mn><mo>\u2218<\/mo><\/msup><annotation encoding=\"application\/x-tex\">48^\\circ<\/annotation><\/semantics><\/math>, the measure of <math data-latex=\"\\angle{SXQ}\"><semantics><mrow><mi>\u2220<\/mi><mrow><mi>S<\/mi><mi>X<\/mi><mi>Q<\/mi><\/mrow><\/mrow><annotation encoding=\"application\/x-tex\">\\angle{SXQ}<\/annotation><\/semantics><\/math> is <math data-latex=\"86^\\circ\"><semantics><msup><mn>86<\/mn><mo>\u2218<\/mo><\/msup><annotation encoding=\"application\/x-tex\">86^\\circ<\/annotation><\/semantics><\/math>, the measure of <math data-latex=\"\\angle{SWU}\"><semantics><mrow><mi>\u2220<\/mi><mrow><mi>S<\/mi><mi>W<\/mi><mi>U<\/mi><\/mrow><\/mrow><annotation encoding=\"application\/x-tex\">\\angle{SWU}<\/annotation><\/semantics><\/math> is <math data-latex=\"85^\\circ\"><semantics><msup><mn>85<\/mn><mo>\u2218<\/mo><\/msup><annotation encoding=\"application\/x-tex\">85^\\circ<\/annotation><\/semantics><\/math>, and the measure of <math data-latex=\"\\angle{VTU}\"><semantics><mrow><mi>\u2220<\/mi><mrow><mi>V<\/mi><mi>T<\/mi><mi>U<\/mi><\/mrow><\/mrow><annotation encoding=\"application\/x-tex\">\\angle{VTU}<\/annotation><\/semantics><\/math> is <math data-latex=\"162^\\circ\"><semantics><msup><mn>162<\/mn><mo>\u2218<\/mo><\/msup><annotation encoding=\"application\/x-tex\">162^\\circ<\/annotation><\/semantics><\/math>. What is the measure, in degrees, of <math data-latex=\"\\angle{TUR}\"><semantics><mrow><mi>\u2220<\/mi><mrow><mi>T<\/mi><mi>U<\/mi><mi>R<\/mi><\/mrow><\/mrow><annotation encoding=\"application\/x-tex\">\\angle{TUR}<\/annotation><\/semantics><\/math>?<br>A) 123<br>B) 134<br>C) 145<br>D) 156<\/p>\n\n\n\n<p class=\"is-style-info\" style=\"font-size:0.9em\">\u2705 <strong>Understand the QUESTION \u2014 Multi-Triangle Angle Chase<\/strong><br><strong>Given<\/strong><br>~ Points <math><semantics><mrow><mi>Q<\/mi><mo separator=\"true\">,<\/mo><mi>R<\/mi><mo separator=\"true\">,<\/mo><mi>S<\/mi><mo separator=\"true\">,<\/mo><mi>T<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">Q, R, S, T<\/annotation><\/semantics><\/math> lie on straight line <math><semantics><mrow><mi>P<\/mi><mi>V<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">PV<\/annotation><\/semantics><\/math><br>~ Lines intersect at <math><semantics><mrow><mi>W<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">W<\/annotation><\/semantics><\/math><br>~ Given angle measures:<br><math><semantics><mrow><mi mathvariant=\"normal\">\u2220<\/mi><mi>S<\/mi><mi>Q<\/mi><mi>X<\/mi><mo>=<\/mo><msup><mn>48<\/mn><mo>\u2218<\/mo><\/msup><\/mrow><annotation encoding=\"application\/x-tex\">\\angle SQX = 48^\\circ<\/annotation><\/semantics><\/math><br><math><semantics><mrow><mi mathvariant=\"normal\">\u2220<\/mi><mi>S<\/mi><mi>X<\/mi><mi>Q<\/mi><mo>=<\/mo><msup><mn>86<\/mn><mo>\u2218<\/mo><\/msup><\/mrow><annotation encoding=\"application\/x-tex\">\\angle SXQ = 86^\\circ<\/annotation><\/semantics><\/math><br><math><semantics><mrow><mi mathvariant=\"normal\">\u2220<\/mi><mi>S<\/mi><mi>W<\/mi><mi>U<\/mi><mo>=<\/mo><msup><mn>85<\/mn><mo>\u2218<\/mo><\/msup><\/mrow><annotation encoding=\"application\/x-tex\">\\angle SWU = 85^\\circ<\/annotation><\/semantics><\/math><br><math><semantics><mrow><mi mathvariant=\"normal\">\u2220<\/mi><mi>V<\/mi><mi>T<\/mi><mi>U<\/mi><mo>=<\/mo><msup><mn>162<\/mn><mo>\u2218<\/mo><\/msup><\/mrow><annotation encoding=\"application\/x-tex\">\\angle VTU = 162^\\circ<\/annotation><\/semantics><\/math><br>Find <math><semantics><mrow><mi mathvariant=\"normal\">\u2220<\/mi><mi>T<\/mi><mi>U<\/mi><mi>R<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">\\angle TUR<\/annotation><\/semantics><\/math><br><br><strong>1\ufe0f\u20e3 Key Geometry Rules<br><\/strong>\ud83d\udccc Triangle Sum Rule<math display=\"block\"><semantics><mrow><mtext>Angles&nbsp;in&nbsp;triangle<\/mtext><mo>=<\/mo><msup><mn>180<\/mn><mo>\u2218<\/mo><\/msup><\/mrow><annotation encoding=\"application\/x-tex\">\\text{Angles in triangle} = 180^\\circ<\/annotation><\/semantics><\/math><br>\ud83d\udccc Linear Pair Rule<br>Angles on a straight line sum to:<math display=\"block\"><semantics><mrow><msup><mn>180<\/mn><mo>\u2218<\/mo><\/msup><\/mrow><annotation encoding=\"application\/x-tex\">180^\\circ<\/annotation><\/semantics><\/math><br>\ud83d\udccc Vertical Angles<br>Vertical angles are <strong>equal<\/strong><\/p>\n\n\n\n<p class=\"is-style-success\" style=\"font-size:0.9em\"><strong>2\ufe0f\u20e3 Step-by-Step Logic<\/strong><br><strong><em>\ud83d\udd39 Step 1: What we have and what we need to find:<\/em><\/strong><br>Find <math data-latex=\"\\angle TUR\"><semantics><mrow><mi>\u2220<\/mi><mi>T<\/mi><mi>U<\/mi><mi>R<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">\\angle TUR<\/annotation><\/semantics><\/math>, to find out <math data-latex=\"\\angle TUR\"><semantics><mrow><mi>\u2220<\/mi><mi>T<\/mi><mi>U<\/mi><mi>R<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">\\angle TUR<\/annotation><\/semantics><\/math>, we need <math data-latex=\"\\angle URT\"><semantics><mrow><mi>\u2220<\/mi><mi>U<\/mi><mi>R<\/mi><mi>T<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">\\angle URT<\/annotation><\/semantics><\/math> and <math data-latex=\"\\angle RTU\"><semantics><mrow><mi>\u2220<\/mi><mi>R<\/mi><mi>T<\/mi><mi>U<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">\\angle RTU<\/annotation><\/semantics><\/math>.<br>~ To find <math data-latex=\"\\angle RTU\"><semantics><mrow><mi>\u2220<\/mi><mi>R<\/mi><mi>T<\/mi><mi>U<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">\\angle RTU<\/annotation><\/semantics><\/math>, we have given <math data-latex=\"\\angle VTU\"><semantics><mrow><mi>\u2220<\/mi><mi>V<\/mi><mi>T<\/mi><mi>U<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">\\angle VTU<\/annotation><\/semantics><\/math>.<br>~ To find <math data-latex=\"\\angle URT\"><semantics><mrow><mi>\u2220<\/mi><mi>U<\/mi><mi>R<\/mi><mi>T<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">\\angle URT<\/annotation><\/semantics><\/math>, we need to find <math data-latex=\"\\angle WRS\"><semantics><mrow><mi>\u2220<\/mi><mi>W<\/mi><mi>R<\/mi><mi>S<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">\\angle WRS<\/annotation><\/semantics><\/math>.<br>~ To find <math data-latex=\"\\angle WRS, \"><semantics><mrow><mi>\u2220<\/mi><mi>W<\/mi><mi>R<\/mi><mi>S<\/mi><mo separator=\"true\">,<\/mo><\/mrow><annotation encoding=\"application\/x-tex\">\\angle WRS, <\/annotation><\/semantics><\/math> we need to find <math data-latex=\"\\angle SWR\"><semantics><mrow><mi>\u2220<\/mi><mi>S<\/mi><mi>W<\/mi><mi>R<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">\\angle SWR<\/annotation><\/semantics><\/math> and <math data-latex=\"\\angle RSW\"><semantics><mrow><mi>\u2220<\/mi><mi>R<\/mi><mi>S<\/mi><mi>W<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">\\angle RSW<\/annotation><\/semantics><\/math>.<br>~ ~ To find <math data-latex=\"\\angle SWR\"><semantics><mrow><mi>\u2220<\/mi><mi>S<\/mi><mi>W<\/mi><mi>R<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">\\angle SWR<\/annotation><\/semantics><\/math>, we have given <math data-latex=\"\\angle SWU\"><semantics><mrow><mi>\u2220<\/mi><mi>S<\/mi><mi>W<\/mi><mi>U<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">\\angle SWU<\/annotation><\/semantics><\/math>.<br>~ ~ To find <math data-latex=\"\\angle RSW\"><semantics><mrow><mi>\u2220<\/mi><mi>R<\/mi><mi>S<\/mi><mi>W<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">\\angle RSW<\/annotation><\/semantics><\/math>, we need to find <math data-latex=\"\\angle QSX\"><semantics><mrow><mi>\u2220<\/mi><mi>Q<\/mi><mi>S<\/mi><mi>X<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">\\angle QSX<\/annotation><\/semantics><\/math>.<br>~ ~ ~ To find <math data-latex=\"\\angle QSX\"><semantics><mrow><mi>\u2220<\/mi><mi>Q<\/mi><mi>S<\/mi><mi>X<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">\\angle QSX<\/annotation><\/semantics><\/math>, we have given <math data-latex=\"\\angle SQX\"><semantics><mrow><mi>\u2220<\/mi><mi>S<\/mi><mi>Q<\/mi><mi>X<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">\\angle SQX<\/annotation><\/semantics><\/math> and <math data-latex=\"\\angle SXQ\"><semantics><mrow><mi>\u2220<\/mi><mi>S<\/mi><mi>X<\/mi><mi>Q<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">\\angle SXQ<\/annotation><\/semantics><\/math>.<br>It looks <strong>difficult<\/strong> but actually, there are only <strong>3-Key rules<\/strong> here. Once you master this type of Questions. All problems from Line, Angle, and Triangles will be <strong>easy<\/strong> for you.<br><br><strong><em>\ud83d\udd39 Step 2: Find <\/em><\/strong><math data-latex=\"\\angle QSX\"><semantics><mrow><mi>\u2220<\/mi><mi>Q<\/mi><mi>S<\/mi><mi>X<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">\\angle QSX<\/annotation><\/semantics><\/math>:<br>We use Triangle Sum Rule because <math data-latex=\"\\angle QSX, \\angle SQX\"><semantics><mrow><mi>\u2220<\/mi><mi>Q<\/mi><mi>S<\/mi><mi>X<\/mi><mo separator=\"true\">,<\/mo><mi>\u2220<\/mi><mi>S<\/mi><mi>Q<\/mi><mi>X<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">\\angle QSX, \\angle SQX<\/annotation><\/semantics><\/math>, and <math data-latex=\"\\angle SXQ\"><semantics><mrow><mi>\u2220<\/mi><mi>S<\/mi><mi>X<\/mi><mi>Q<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">\\angle SXQ<\/annotation><\/semantics><\/math> are angles of one triangle,<br>So<br><math data-latex=\"Angles\\ in\\ Triangle = 180^\\circ\\\\ \\\\ \\angle QSX + \\angle SQX + \\angle SXQ = 180^\\circ\\\\ \\\\ \\angle QSX + 48^\\circ + 86^\\circ = 180^\\circ \\\\ \\\\ \\angle QSX + 134^\\circ = 180^\\circ \\\\ \\\\ \\angle QSX = 180^\\circ - 134^\\circ \\\\ \\\\ \\angle QSX = 46^\\circ\"><semantics><mtable columnalign=\"left\" rowspacing=\"0em\"><mtr><mtd style=\"text-align:left\"><mrow><mi>A<\/mi><mi>n<\/mi><mi>g<\/mi><mi>l<\/mi><mi>e<\/mi><mi>s<\/mi><mtext>&nbsp;<\/mtext><mi>i<\/mi><mi>n<\/mi><mtext>&nbsp;<\/mtext><mi>T<\/mi><mi>r<\/mi><mi>i<\/mi><mi>a<\/mi><mi>n<\/mi><mi>g<\/mi><mi>l<\/mi><mi>e<\/mi><mo>=<\/mo><msup><mn>180<\/mn><mo>\u2218<\/mo><\/msup><\/mrow><mo><\/mo><\/mtd><\/mtr><mtr><mtd style=\"text-align:left\"><mo><\/mo><\/mtd><\/mtr><mtr><mtd style=\"text-align:left\"><mrow><mi>\u2220<\/mi><mi>Q<\/mi><mi>S<\/mi><mi>X<\/mi><mo>+<\/mo><mi>\u2220<\/mi><mi>S<\/mi><mi>Q<\/mi><mi>X<\/mi><mo>+<\/mo><mi>\u2220<\/mi><mi>S<\/mi><mi>X<\/mi><mi>Q<\/mi><mo>=<\/mo><msup><mn>180<\/mn><mo>\u2218<\/mo><\/msup><\/mrow><mo><\/mo><\/mtd><\/mtr><mtr><mtd style=\"text-align:left\"><mo><\/mo><\/mtd><\/mtr><mtr><mtd style=\"text-align:left\"><mrow><mi>\u2220<\/mi><mi>Q<\/mi><mi>S<\/mi><mi>X<\/mi><mo>+<\/mo><msup><mn>48<\/mn><mo>\u2218<\/mo><\/msup><mo>+<\/mo><msup><mn>86<\/mn><mo>\u2218<\/mo><\/msup><mo>=<\/mo><msup><mn>180<\/mn><mo>\u2218<\/mo><\/msup><\/mrow><mo><\/mo><\/mtd><\/mtr><mtr><mtd style=\"text-align:left\"><mo><\/mo><\/mtd><\/mtr><mtr><mtd style=\"text-align:left\"><mrow><mi>\u2220<\/mi><mi>Q<\/mi><mi>S<\/mi><mi>X<\/mi><mo>+<\/mo><msup><mn>134<\/mn><mo>\u2218<\/mo><\/msup><mo>=<\/mo><msup><mn>180<\/mn><mo>\u2218<\/mo><\/msup><\/mrow><mo><\/mo><\/mtd><\/mtr><mtr><mtd style=\"text-align:left\"><mo><\/mo><\/mtd><\/mtr><mtr><mtd style=\"text-align:left\"><mrow><mi>\u2220<\/mi><mi>Q<\/mi><mi>S<\/mi><mi>X<\/mi><mo>=<\/mo><msup><mn>180<\/mn><mo>\u2218<\/mo><\/msup><mo>\u2212<\/mo><msup><mn>134<\/mn><mo>\u2218<\/mo><\/msup><\/mrow><mo><\/mo><\/mtd><\/mtr><mtr><mtd style=\"text-align:left\"><mo><\/mo><\/mtd><\/mtr><mtr><mtd style=\"text-align:left\"><mrow><mi>\u2220<\/mi><mi>Q<\/mi><mi>S<\/mi><mi>X<\/mi><mo>=<\/mo><msup><mn>46<\/mn><mo>\u2218<\/mo><\/msup><\/mrow><\/mtd><\/mtr><\/mtable><annotation encoding=\"application\/x-tex\">Angles\\ in\\ Triangle = 180^\\circ\\\\ \\\\ \\angle QSX + \\angle SQX + \\angle SXQ = 180^\\circ\\\\ \\\\ \\angle QSX + 48^\\circ + 86^\\circ = 180^\\circ \\\\ \\\\ \\angle QSX + 134^\\circ = 180^\\circ \\\\ \\\\ \\angle QSX = 180^\\circ &#8211; 134^\\circ \\\\ \\\\ \\angle QSX = 46^\\circ<\/annotation><\/semantics><\/math><br><br><strong><em>\ud83d\udd39 Step 3: Find <\/em><\/strong><math data-latex=\"\\angle RSW\"><semantics><mrow><mi>\u2220<\/mi><mi>R<\/mi><mi>S<\/mi><mi>W<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">\\angle RSW<\/annotation><\/semantics><\/math>:<br>Look at the figure, notice that <math data-latex=\"\\angle RSW\"><semantics><mrow><mi>\u2220<\/mi><mi>R<\/mi><mi>S<\/mi><mi>W<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">\\angle RSW<\/annotation><\/semantics><\/math> and <math data-latex=\"\\angle QSX\"><semantics><mrow><mi>\u2220<\/mi><mi>Q<\/mi><mi>S<\/mi><mi>X<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">\\angle QSX<\/annotation><\/semantics><\/math> are on the same vertical angle. Look at the point <math data-latex=\"S\"><semantics><mi>S<\/mi><annotation encoding=\"application\/x-tex\">S<\/annotation><\/semantics><\/math> on both,<br>So<br><math data-latex=\"Vertical\\ angles\\ are\\ equal.\"><semantics><mrow><mi>V<\/mi><mi>e<\/mi><mi>r<\/mi><mi>t<\/mi><mi>i<\/mi><mi>c<\/mi><mi>a<\/mi><mi>l<\/mi><mtext>&nbsp;<\/mtext><mi>a<\/mi><mi>n<\/mi><mi>g<\/mi><mi>l<\/mi><mi>e<\/mi><mi>s<\/mi><mtext>&nbsp;<\/mtext><mi>a<\/mi><mi>r<\/mi><mi>e<\/mi><mtext>&nbsp;<\/mtext><mi>e<\/mi><mi>q<\/mi><mi>u<\/mi><mi>a<\/mi><mi>l<\/mi><mi>.<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">Vertical\\ angles\\ are\\ equal.<\/annotation><\/semantics><\/math><br><math data-latex=\"\\angle RSW = \\angle QSX \\\\ \\\\ \\angle RSW = 46^\\circ\"><semantics><mtable columnalign=\"left\" rowspacing=\"0em\"><mtr><mtd style=\"text-align:left\"><mrow><mi>\u2220<\/mi><mi>R<\/mi><mi>S<\/mi><mi>W<\/mi><mo>=<\/mo><mi>\u2220<\/mi><mi>Q<\/mi><mi>S<\/mi><mi>X<\/mi><\/mrow><mo><\/mo><\/mtd><\/mtr><mtr><mtd style=\"text-align:left\"><mo><\/mo><\/mtd><\/mtr><mtr><mtd style=\"text-align:left\"><mrow><mi>\u2220<\/mi><mi>R<\/mi><mi>S<\/mi><mi>W<\/mi><mo>=<\/mo><msup><mn>46<\/mn><mo>\u2218<\/mo><\/msup><\/mrow><\/mtd><\/mtr><\/mtable><annotation encoding=\"application\/x-tex\">\\angle RSW = \\angle QSX \\\\ \\\\ \\angle RSW = 46^\\circ<\/annotation><\/semantics><\/math><br><br><strong><em>\ud83d\udd39 Step 4: Find<\/em><\/strong> <math data-latex=\"\\angle SWR\"><semantics><mrow><mi>\u2220<\/mi><mi>S<\/mi><mi>W<\/mi><mi>R<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">\\angle SWR<\/annotation><\/semantics><\/math>:<br>Notice on the figure that <math data-latex=\"\\angle SWR\"><semantics><mrow><mi>\u2220<\/mi><mi>S<\/mi><mi>W<\/mi><mi>R<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">\\angle SWR<\/annotation><\/semantics><\/math> and <math data-latex=\"\\angle SWU\"><semantics><mrow><mi>\u2220<\/mi><mi>S<\/mi><mi>W<\/mi><mi>U<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">\\angle SWU<\/annotation><\/semantics><\/math> are a linear pair. Look at the points <math data-latex=\"S\"><semantics><mi>S<\/mi><annotation encoding=\"application\/x-tex\">S<\/annotation><\/semantics><\/math> and <math data-latex=\"W\"><semantics><mi>W<\/mi><annotation encoding=\"application\/x-tex\">W<\/annotation><\/semantics><\/math>,<br>So<br><math data-latex=\"Angle\\ on\\ a\\ straight\\ line\\ sum\\ to\\ 180^\\circ\\\\ \\\\ \\angle SWR + \\angle SWU = 180^\\circ \\\\ \\\\ \\angle SWR + 85^\\circ = 180^\\circ \\\\ \\\\ \\angle SWR = 180^\\circ - 85^\\circ \\\\ \\\\ \\angle SWR = 95^\\circ\"><semantics><mtable columnalign=\"left\" rowspacing=\"0em\"><mtr><mtd style=\"text-align:left\"><mrow><mi>A<\/mi><mi>n<\/mi><mi>g<\/mi><mi>l<\/mi><mi>e<\/mi><mtext>&nbsp;<\/mtext><mi>o<\/mi><mi>n<\/mi><mtext>&nbsp;<\/mtext><mi>a<\/mi><mtext>&nbsp;<\/mtext><mi>s<\/mi><mi>t<\/mi><mi>r<\/mi><mi>a<\/mi><mi>i<\/mi><mi>g<\/mi><mi>h<\/mi><mi>t<\/mi><mtext>&nbsp;<\/mtext><mi>l<\/mi><mi>i<\/mi><mi>n<\/mi><mi>e<\/mi><mtext>&nbsp;<\/mtext><mi>s<\/mi><mi>u<\/mi><mi>m<\/mi><mtext>&nbsp;<\/mtext><mi>t<\/mi><mi>o<\/mi><mtext>&nbsp;<\/mtext><msup><mn>180<\/mn><mo>\u2218<\/mo><\/msup><\/mrow><mo><\/mo><\/mtd><\/mtr><mtr><mtd style=\"text-align:left\"><mo><\/mo><\/mtd><\/mtr><mtr><mtd style=\"text-align:left\"><mrow><mi>\u2220<\/mi><mi>S<\/mi><mi>W<\/mi><mi>R<\/mi><mo>+<\/mo><mi>\u2220<\/mi><mi>S<\/mi><mi>W<\/mi><mi>U<\/mi><mo>=<\/mo><msup><mn>180<\/mn><mo>\u2218<\/mo><\/msup><\/mrow><mo><\/mo><\/mtd><\/mtr><mtr><mtd style=\"text-align:left\"><mo><\/mo><\/mtd><\/mtr><mtr><mtd style=\"text-align:left\"><mrow><mi>\u2220<\/mi><mi>S<\/mi><mi>W<\/mi><mi>R<\/mi><mo>+<\/mo><msup><mn>85<\/mn><mo>\u2218<\/mo><\/msup><mo>=<\/mo><msup><mn>180<\/mn><mo>\u2218<\/mo><\/msup><\/mrow><mo><\/mo><\/mtd><\/mtr><mtr><mtd style=\"text-align:left\"><mo><\/mo><\/mtd><\/mtr><mtr><mtd style=\"text-align:left\"><mrow><mi>\u2220<\/mi><mi>S<\/mi><mi>W<\/mi><mi>R<\/mi><mo>=<\/mo><msup><mn>180<\/mn><mo>\u2218<\/mo><\/msup><mo>\u2212<\/mo><msup><mn>85<\/mn><mo>\u2218<\/mo><\/msup><\/mrow><mo><\/mo><\/mtd><\/mtr><mtr><mtd style=\"text-align:left\"><mo><\/mo><\/mtd><\/mtr><mtr><mtd style=\"text-align:left\"><mrow><mi>\u2220<\/mi><mi>S<\/mi><mi>W<\/mi><mi>R<\/mi><mo>=<\/mo><msup><mn>95<\/mn><mo>\u2218<\/mo><\/msup><\/mrow><\/mtd><\/mtr><\/mtable><annotation encoding=\"application\/x-tex\">Angle\\ on\\ a\\ straight\\ line\\ sum\\ to\\ 180^\\circ\\\\ \\\\ \\angle SWR + \\angle SWU = 180^\\circ \\\\ \\\\ \\angle SWR + 85^\\circ = 180^\\circ \\\\ \\\\ \\angle SWR = 180^\\circ &#8211; 85^\\circ \\\\ \\\\ \\angle SWR = 95^\\circ<\/annotation><\/semantics><\/math><br><br><strong><em>\ud83d\udd39 Step 5: Find <\/em><\/strong><math data-latex=\"\\angle WRS\"><semantics><mrow><mi>\u2220<\/mi><mi>W<\/mi><mi>R<\/mi><mi>S<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">\\angle WRS<\/annotation><\/semantics><\/math>:<br>We use Triangle Sum Rule because <math data-latex=\"\\angle WRS,\\ \\angle SWR\"><semantics><mrow><mi>\u2220<\/mi><mi>W<\/mi><mi>R<\/mi><mi>S<\/mi><mo separator=\"true\">,<\/mo><mtext>&nbsp;<\/mtext><mi>\u2220<\/mi><mi>S<\/mi><mi>W<\/mi><mi>R<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">\\angle WRS,\\ \\angle SWR<\/annotation><\/semantics><\/math>, and <math data-latex=\"\\angle RSW\"><semantics><mrow><mi>\u2220<\/mi><mi>R<\/mi><mi>S<\/mi><mi>W<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">\\angle RSW<\/annotation><\/semantics><\/math> are angles of one triangle,<br>So<br><math data-latex=\"Angles\\ in\\ Triangle = 180^\\circ\\\\ \\\\ \\angle WRS + \\angle SWR + \\angle RSW = 180^\\circ\\\\ \\\\ \\angle WRS + 95^\\circ + 46^\\circ = 180^\\circ \\\\ \\\\ \\angle WRS + 141^\\circ = 180^\\circ \\\\ \\\\ \\angle WRS = 180^\\circ - 141^\\circ \\\\ \\\\ \\angle WRS = 39^\\circ\"><semantics><mtable columnalign=\"left\" rowspacing=\"0em\"><mtr><mtd style=\"text-align:left\"><mrow><mi>A<\/mi><mi>n<\/mi><mi>g<\/mi><mi>l<\/mi><mi>e<\/mi><mi>s<\/mi><mtext>&nbsp;<\/mtext><mi>i<\/mi><mi>n<\/mi><mtext>&nbsp;<\/mtext><mi>T<\/mi><mi>r<\/mi><mi>i<\/mi><mi>a<\/mi><mi>n<\/mi><mi>g<\/mi><mi>l<\/mi><mi>e<\/mi><mo>=<\/mo><msup><mn>180<\/mn><mo>\u2218<\/mo><\/msup><\/mrow><mo><\/mo><\/mtd><\/mtr><mtr><mtd style=\"text-align:left\"><mo><\/mo><\/mtd><\/mtr><mtr><mtd style=\"text-align:left\"><mrow><mi>\u2220<\/mi><mi>W<\/mi><mi>R<\/mi><mi>S<\/mi><mo>+<\/mo><mi>\u2220<\/mi><mi>S<\/mi><mi>W<\/mi><mi>R<\/mi><mo>+<\/mo><mi>\u2220<\/mi><mi>R<\/mi><mi>S<\/mi><mi>W<\/mi><mo>=<\/mo><msup><mn>180<\/mn><mo>\u2218<\/mo><\/msup><\/mrow><mo><\/mo><\/mtd><\/mtr><mtr><mtd style=\"text-align:left\"><mo><\/mo><\/mtd><\/mtr><mtr><mtd style=\"text-align:left\"><mrow><mi>\u2220<\/mi><mi>W<\/mi><mi>R<\/mi><mi>S<\/mi><mo>+<\/mo><msup><mn>95<\/mn><mo>\u2218<\/mo><\/msup><mo>+<\/mo><msup><mn>46<\/mn><mo>\u2218<\/mo><\/msup><mo>=<\/mo><msup><mn>180<\/mn><mo>\u2218<\/mo><\/msup><\/mrow><mo><\/mo><\/mtd><\/mtr><mtr><mtd style=\"text-align:left\"><mo><\/mo><\/mtd><\/mtr><mtr><mtd style=\"text-align:left\"><mrow><mi>\u2220<\/mi><mi>W<\/mi><mi>R<\/mi><mi>S<\/mi><mo>+<\/mo><msup><mn>141<\/mn><mo>\u2218<\/mo><\/msup><mo>=<\/mo><msup><mn>180<\/mn><mo>\u2218<\/mo><\/msup><\/mrow><mo><\/mo><\/mtd><\/mtr><mtr><mtd style=\"text-align:left\"><mo><\/mo><\/mtd><\/mtr><mtr><mtd style=\"text-align:left\"><mrow><mi>\u2220<\/mi><mi>W<\/mi><mi>R<\/mi><mi>S<\/mi><mo>=<\/mo><msup><mn>180<\/mn><mo>\u2218<\/mo><\/msup><mo>\u2212<\/mo><msup><mn>141<\/mn><mo>\u2218<\/mo><\/msup><\/mrow><mo><\/mo><\/mtd><\/mtr><mtr><mtd style=\"text-align:left\"><mo><\/mo><\/mtd><\/mtr><mtr><mtd style=\"text-align:left\"><mrow><mi>\u2220<\/mi><mi>W<\/mi><mi>R<\/mi><mi>S<\/mi><mo>=<\/mo><msup><mn>39<\/mn><mo>\u2218<\/mo><\/msup><\/mrow><\/mtd><\/mtr><\/mtable><annotation encoding=\"application\/x-tex\">Angles\\ in\\ Triangle = 180^\\circ\\\\ \\\\ \\angle WRS + \\angle SWR + \\angle RSW = 180^\\circ\\\\ \\\\ \\angle WRS + 95^\\circ + 46^\\circ = 180^\\circ \\\\ \\\\ \\angle WRS + 141^\\circ = 180^\\circ \\\\ \\\\ \\angle WRS = 180^\\circ &#8211; 141^\\circ \\\\ \\\\ \\angle WRS = 39^\\circ<\/annotation><\/semantics><\/math><br><br><strong><em>\ud83d\udd39 Step 6: Find <\/em><\/strong><math data-latex=\"\\angle URT\"><semantics><mrow><mi>\u2220<\/mi><mi>U<\/mi><mi>R<\/mi><mi>T<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">\\angle URT<\/annotation><\/semantics><\/math>:<br>Look at the figure, notice that <math data-latex=\"\\angle URT\"><semantics><mrow><mi>\u2220<\/mi><mi>U<\/mi><mi>R<\/mi><mi>T<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">\\angle URT<\/annotation><\/semantics><\/math> and <math data-latex=\"\\angle WRS\"><semantics><mrow><mi>\u2220<\/mi><mi>W<\/mi><mi>R<\/mi><mi>S<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">\\angle WRS<\/annotation><\/semantics><\/math> are on the same vertical angle. Look at the point <math data-latex=\"R\"><semantics><mi>R<\/mi><annotation encoding=\"application\/x-tex\">R<\/annotation><\/semantics><\/math> on both,<br>So<br><math data-latex=\"Vertical\\ angles\\ are\\ equal.\"><semantics><mrow><mi>V<\/mi><mi>e<\/mi><mi>r<\/mi><mi>t<\/mi><mi>i<\/mi><mi>c<\/mi><mi>a<\/mi><mi>l<\/mi><mtext>&nbsp;<\/mtext><mi>a<\/mi><mi>n<\/mi><mi>g<\/mi><mi>l<\/mi><mi>e<\/mi><mi>s<\/mi><mtext>&nbsp;<\/mtext><mi>a<\/mi><mi>r<\/mi><mi>e<\/mi><mtext>&nbsp;<\/mtext><mi>e<\/mi><mi>q<\/mi><mi>u<\/mi><mi>a<\/mi><mi>l<\/mi><mi>.<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">Vertical\\ angles\\ are\\ equal.<\/annotation><\/semantics><\/math><br><math data-latex=\"\\angle URT = \\angle WRS \\\\ \\\\ \\angle URT = 39^\\circ\"><semantics><mtable columnalign=\"left\" rowspacing=\"0em\"><mtr><mtd style=\"text-align:left\"><mrow><mi>\u2220<\/mi><mi>U<\/mi><mi>R<\/mi><mi>T<\/mi><mo>=<\/mo><mi>\u2220<\/mi><mi>W<\/mi><mi>R<\/mi><mi>S<\/mi><\/mrow><mo><\/mo><\/mtd><\/mtr><mtr><mtd style=\"text-align:left\"><mo><\/mo><\/mtd><\/mtr><mtr><mtd style=\"text-align:left\"><mrow><mi>\u2220<\/mi><mi>U<\/mi><mi>R<\/mi><mi>T<\/mi><mo>=<\/mo><msup><mn>39<\/mn><mo>\u2218<\/mo><\/msup><\/mrow><\/mtd><\/mtr><\/mtable><annotation encoding=\"application\/x-tex\">\\angle URT = \\angle WRS \\\\ \\\\ \\angle URT = 39^\\circ<\/annotation><\/semantics><\/math><br><br><strong><em>\ud83d\udd39 Step 7: Find<\/em><\/strong> <math data-latex=\"\\angle RTU\"><semantics><mrow><mi>\u2220<\/mi><mi>R<\/mi><mi>T<\/mi><mi>U<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">\\angle RTU<\/annotation><\/semantics><\/math>:<br>Notice on the figure that <math data-latex=\"\\angle RTU\"><semantics><mrow><mi>\u2220<\/mi><mi>R<\/mi><mi>T<\/mi><mi>U<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">\\angle RTU<\/annotation><\/semantics><\/math> and <math data-latex=\"\\angle VTU\"><semantics><mrow><mi>\u2220<\/mi><mi>V<\/mi><mi>T<\/mi><mi>U<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">\\angle VTU<\/annotation><\/semantics><\/math> are a linear pair. Look at the points <math data-latex=\"T\"><semantics><mi>T<\/mi><annotation encoding=\"application\/x-tex\">T<\/annotation><\/semantics><\/math> and <math data-latex=\"U\"><semantics><mi>U<\/mi><annotation encoding=\"application\/x-tex\">U<\/annotation><\/semantics><\/math>,<br>So<br><math data-latex=\"Angle\\ on\\ a\\ straight\\ line\\ sum\\ to\\ 180^\\circ\\\\ \\\\ \\angle RTU + \\angle VTU = 180^\\circ \\\\ \\\\ \\angle RTU + 162^\\circ = 180^\\circ \\\\ \\\\ \\angle RTU = 180^\\circ - 162^\\circ \\\\ \\\\ \\angle RTU = 18^\\circ\"><semantics><mtable columnalign=\"left\" rowspacing=\"0em\"><mtr><mtd style=\"text-align:left\"><mrow><mi>A<\/mi><mi>n<\/mi><mi>g<\/mi><mi>l<\/mi><mi>e<\/mi><mtext>&nbsp;<\/mtext><mi>o<\/mi><mi>n<\/mi><mtext>&nbsp;<\/mtext><mi>a<\/mi><mtext>&nbsp;<\/mtext><mi>s<\/mi><mi>t<\/mi><mi>r<\/mi><mi>a<\/mi><mi>i<\/mi><mi>g<\/mi><mi>h<\/mi><mi>t<\/mi><mtext>&nbsp;<\/mtext><mi>l<\/mi><mi>i<\/mi><mi>n<\/mi><mi>e<\/mi><mtext>&nbsp;<\/mtext><mi>s<\/mi><mi>u<\/mi><mi>m<\/mi><mtext>&nbsp;<\/mtext><mi>t<\/mi><mi>o<\/mi><mtext>&nbsp;<\/mtext><msup><mn>180<\/mn><mo>\u2218<\/mo><\/msup><\/mrow><mo><\/mo><\/mtd><\/mtr><mtr><mtd style=\"text-align:left\"><mo><\/mo><\/mtd><\/mtr><mtr><mtd style=\"text-align:left\"><mrow><mi>\u2220<\/mi><mi>R<\/mi><mi>T<\/mi><mi>U<\/mi><mo>+<\/mo><mi>\u2220<\/mi><mi>V<\/mi><mi>T<\/mi><mi>U<\/mi><mo>=<\/mo><msup><mn>180<\/mn><mo>\u2218<\/mo><\/msup><\/mrow><mo><\/mo><\/mtd><\/mtr><mtr><mtd style=\"text-align:left\"><mo><\/mo><\/mtd><\/mtr><mtr><mtd style=\"text-align:left\"><mrow><mi>\u2220<\/mi><mi>R<\/mi><mi>T<\/mi><mi>U<\/mi><mo>+<\/mo><msup><mn>162<\/mn><mo>\u2218<\/mo><\/msup><mo>=<\/mo><msup><mn>180<\/mn><mo>\u2218<\/mo><\/msup><\/mrow><mo><\/mo><\/mtd><\/mtr><mtr><mtd style=\"text-align:left\"><mo><\/mo><\/mtd><\/mtr><mtr><mtd style=\"text-align:left\"><mrow><mi>\u2220<\/mi><mi>R<\/mi><mi>T<\/mi><mi>U<\/mi><mo>=<\/mo><msup><mn>180<\/mn><mo>\u2218<\/mo><\/msup><mo>\u2212<\/mo><msup><mn>162<\/mn><mo>\u2218<\/mo><\/msup><\/mrow><mo><\/mo><\/mtd><\/mtr><mtr><mtd style=\"text-align:left\"><mo><\/mo><\/mtd><\/mtr><mtr><mtd style=\"text-align:left\"><mrow><mi>\u2220<\/mi><mi>R<\/mi><mi>T<\/mi><mi>U<\/mi><mo>=<\/mo><msup><mn>18<\/mn><mo>\u2218<\/mo><\/msup><\/mrow><\/mtd><\/mtr><\/mtable><annotation encoding=\"application\/x-tex\">Angle\\ on\\ a\\ straight\\ line\\ sum\\ to\\ 180^\\circ\\\\ \\\\ \\angle RTU + \\angle VTU = 180^\\circ \\\\ \\\\ \\angle RTU + 162^\\circ = 180^\\circ \\\\ \\\\ \\angle RTU = 180^\\circ &#8211; 162^\\circ \\\\ \\\\ \\angle RTU = 18^\\circ<\/annotation><\/semantics><\/math><br><br><strong><em>\ud83d\udd39 Step 8: The Final Part &#8211; Find <\/em><\/strong><math data-latex=\"\\angle TUR\"><semantics><mrow><mi>\u2220<\/mi><mi>T<\/mi><mi>U<\/mi><mi>R<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">\\angle TUR<\/annotation><\/semantics><\/math>:<br>We use Triangle Sum Rule because <math data-latex=\"\\angle TUR, \\angle URT\"><semantics><mrow><mi>\u2220<\/mi><mi>T<\/mi><mi>U<\/mi><mi>R<\/mi><mo separator=\"true\">,<\/mo><mi>\u2220<\/mi><mi>U<\/mi><mi>R<\/mi><mi>T<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">\\angle TUR, \\angle URT<\/annotation><\/semantics><\/math>, and <math data-latex=\"\\angle RTU\"><semantics><mrow><mi>\u2220<\/mi><mi>R<\/mi><mi>T<\/mi><mi>U<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">\\angle RTU<\/annotation><\/semantics><\/math> are angles of one triangle,<br>So<br><math data-latex=\"Angles\\ in\\ Triangle = 180^\\circ\\\\ \\\\ \\angle TUR + \\angle URT + \\angle RTU = 180^\\circ\\\\ \\\\ \\angle TUR + 39^\\circ + 18^\\circ = 180^\\circ \\\\ \\\\ \\angle TUR + 57^\\circ = 180^\\circ \\\\ \\\\ \\angle TUR = 180^\\circ - 57^\\circ \\\\ \\\\ \\angle QSX = 123^\\circ\"><semantics><mtable columnalign=\"left\" rowspacing=\"0em\"><mtr><mtd style=\"text-align:left\"><mrow><mi>A<\/mi><mi>n<\/mi><mi>g<\/mi><mi>l<\/mi><mi>e<\/mi><mi>s<\/mi><mtext>&nbsp;<\/mtext><mi>i<\/mi><mi>n<\/mi><mtext>&nbsp;<\/mtext><mi>T<\/mi><mi>r<\/mi><mi>i<\/mi><mi>a<\/mi><mi>n<\/mi><mi>g<\/mi><mi>l<\/mi><mi>e<\/mi><mo>=<\/mo><msup><mn>180<\/mn><mo>\u2218<\/mo><\/msup><\/mrow><mo><\/mo><\/mtd><\/mtr><mtr><mtd style=\"text-align:left\"><mo><\/mo><\/mtd><\/mtr><mtr><mtd style=\"text-align:left\"><mrow><mi>\u2220<\/mi><mi>T<\/mi><mi>U<\/mi><mi>R<\/mi><mo>+<\/mo><mi>\u2220<\/mi><mi>U<\/mi><mi>R<\/mi><mi>T<\/mi><mo>+<\/mo><mi>\u2220<\/mi><mi>R<\/mi><mi>T<\/mi><mi>U<\/mi><mo>=<\/mo><msup><mn>180<\/mn><mo>\u2218<\/mo><\/msup><\/mrow><mo><\/mo><\/mtd><\/mtr><mtr><mtd style=\"text-align:left\"><mo><\/mo><\/mtd><\/mtr><mtr><mtd style=\"text-align:left\"><mrow><mi>\u2220<\/mi><mi>T<\/mi><mi>U<\/mi><mi>R<\/mi><mo>+<\/mo><msup><mn>39<\/mn><mo>\u2218<\/mo><\/msup><mo>+<\/mo><msup><mn>18<\/mn><mo>\u2218<\/mo><\/msup><mo>=<\/mo><msup><mn>180<\/mn><mo>\u2218<\/mo><\/msup><\/mrow><mo><\/mo><\/mtd><\/mtr><mtr><mtd style=\"text-align:left\"><mo><\/mo><\/mtd><\/mtr><mtr><mtd style=\"text-align:left\"><mrow><mi>\u2220<\/mi><mi>T<\/mi><mi>U<\/mi><mi>R<\/mi><mo>+<\/mo><msup><mn>57<\/mn><mo>\u2218<\/mo><\/msup><mo>=<\/mo><msup><mn>180<\/mn><mo>\u2218<\/mo><\/msup><\/mrow><mo><\/mo><\/mtd><\/mtr><mtr><mtd style=\"text-align:left\"><mo><\/mo><\/mtd><\/mtr><mtr><mtd style=\"text-align:left\"><mrow><mi>\u2220<\/mi><mi>T<\/mi><mi>U<\/mi><mi>R<\/mi><mo>=<\/mo><msup><mn>180<\/mn><mo>\u2218<\/mo><\/msup><mo>\u2212<\/mo><msup><mn>57<\/mn><mo>\u2218<\/mo><\/msup><\/mrow><mo><\/mo><\/mtd><\/mtr><mtr><mtd style=\"text-align:left\"><mo><\/mo><\/mtd><\/mtr><mtr><mtd style=\"text-align:left\"><mrow><mi>\u2220<\/mi><mi>Q<\/mi><mi>S<\/mi><mi>X<\/mi><mo>=<\/mo><msup><mn>123<\/mn><mo>\u2218<\/mo><\/msup><\/mrow><\/mtd><\/mtr><\/mtable><annotation encoding=\"application\/x-tex\">Angles\\ in\\ Triangle = 180^\\circ\\\\ \\\\ \\angle TUR + \\angle URT + \\angle RTU = 180^\\circ\\\\ \\\\ \\angle TUR + 39^\\circ + 18^\\circ = 180^\\circ \\\\ \\\\ \\angle TUR + 57^\\circ = 180^\\circ \\\\ \\\\ \\angle TUR = 180^\\circ &#8211; 57^\\circ \\\\ \\\\ \\angle QSX = 123^\\circ<\/annotation><\/semantics><\/math><br><br>\u2705 <strong>Correct Answer is 123: Option A<\/strong><\/p>\n\n\n\n<p class=\"is-style-error\" style=\"font-size:0.9em\">\u274c <strong>134 \/ 145<\/strong> \u2192 missing vertical angle transfer<br>\u274c <strong>156<\/strong> \u2192 assuming straight-line angle instead of triangle sum<\/p>\n\n\n\n<p class=\"is-style-info\" style=\"font-size:0.9em\"><strong>\ud83e\uddee Desmos?<\/strong><br>\ud83d\udeab Not suitable \u2014 requires <strong>angle inference<\/strong>, not coordinates.<\/p>\n<\/div><\/details><\/div>\n\n\n\n<div class=\"wp-block-coblocks-accordion-item\"><details><summary class=\"wp-block-coblocks-accordion-item__title\"><strong>15th Question<\/strong><\/summary><div class=\"wp-block-coblocks-accordion-item__content\">\n<p class=\"is-style-warning\" style=\"font-size:0.9em\"><strong>Question:<\/strong> A certain apprentice has enrolled in 85 hours of training courses. The equation 10<em>x<\/em> + 15<em>y<\/em> = 85 represents this situation, where <em>x<\/em> is the number of on-site training courses and <em>y<\/em> is the number of online training courses this apprentice has enrolled in. How many more hours does each online training course take than each on-site training course?<br><br>[Type-Based Answer: In the final exam, you will type the answer rather than choose from options.]<\/p>\n\n\n\n<p class=\"is-style-success\" style=\"font-size:0.9em\"><strong>\ud83e\uddee Step-by-Step Solution<br><\/strong>This equation represents <strong>total training hours<\/strong>, not the number of courses.<br>Each term follows the structure:<br><math display=\"block\"><semantics><mrow><mo stretchy=\"false\">(<\/mo><mtext>hours&nbsp;per&nbsp;course<\/mtext><mo stretchy=\"false\">)<\/mo><mo>\u00d7<\/mo><mo stretchy=\"false\">(<\/mo><mtext>number&nbsp;of&nbsp;courses<\/mtext><mo stretchy=\"false\">)<\/mo><\/mrow><annotation encoding=\"application\/x-tex\">(\\text{hours per course}) \\times (\\text{number of courses})<\/annotation><\/semantics><\/math><br>From the equation: 10<em>x<\/em> + 15<em>y<\/em> = 85<br>~ <em>x<\/em> represents on-site course<br>~ 10<em>x<\/em> \u2192 each <strong>on-site<\/strong> course takes <strong>10 hours<\/strong><br>~ <em>y<\/em> represents online course<br>~ 15<em>y<\/em> \u2192 each <strong>online<\/strong> course takes <strong>15 hours<\/strong><br>The question does <strong>not<\/strong> ask for <math><semantics><mrow><mi>x<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">x<\/annotation><\/semantics><\/math> or <math><semantics><mrow><mi>y<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">y<\/annotation><\/semantics><\/math>.<br>It asks for the <strong>difference in hours per course<\/strong>. <em>(Key word: How many <strong>more<\/strong>)<\/em><br>So we compare the coefficients directly.<br><br><strong>Step: Compare course durations<\/strong><br><math display=\"block\"><semantics><mrow><mn>15<\/mn><mo>\u2212<\/mo><mn>10<\/mn><mo>=<\/mo><mn>5<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">15 &#8211; 10 = 5<\/annotation><\/semantics><\/math><br>(A real trap in question if you do not understand or read it properly. Those who miss that try to solve the equation but that is not what the question asks here. It asks:<br><strong><em>How many more hours does each online training course take than each on-site training course?<\/em><\/strong><br>~ It did not ask &#8220;How many more number of courses?&#8221;<br>~ It asks &#8220;How many more hours of online training course?&#8221;<br><br>\u2705 Correct Answer: 5 hours<\/p>\n\n\n\n<p class=\"is-style-error\" style=\"font-size:0.9em\"><strong>\u274c Trap 1: Answer = 85<br>Why students choose this:<\/strong><br>They confuse <strong>total hours<\/strong> with <strong>difference per course<\/strong>.<\/p>\n\n\n\n<p class=\"is-style-error\" style=\"font-size:0.9em\"><strong>\u274c Trap 2: Answer = 25<br>Why students choose this:<\/strong><br>They add the coefficients instead of comparing them:<br><math display=\"block\"><semantics><mrow><mn>10<\/mn><mo>+<\/mo><mn>15<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">10 + 15<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<p class=\"is-style-error\" style=\"font-size:0.9em\"><strong>\u274c Trap 3: Answer = 1.5<\/strong><br><strong>Why students choose this:<\/strong><br>They divide coefficients instead of subtracting:<math display=\"block\"><semantics><mrow><mfrac><mn>15<\/mn><mn>10<\/mn><\/mfrac><\/mrow><annotation encoding=\"application\/x-tex\">\\frac{15}{10}<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<p class=\"is-style-info\" style=\"font-size:0.9em\"><strong>\ud83e\uddee DESMOS CONFIRMATION<\/strong><br>1. Open <strong>Desmos<\/strong><br>2. Type: 10x + 15y = 85<br>3. Observe coefficients directly \u2014 they represent <strong>hours per course<\/strong><br>\u2714 Confirms difference is <strong>5 hours<\/strong><\/p>\n<\/div><\/details><\/div>\n\n\n\n<div class=\"wp-block-coblocks-accordion-item\"><details><summary class=\"wp-block-coblocks-accordion-item__title\"><strong>16th Question<\/strong><\/summary><div class=\"wp-block-coblocks-accordion-item__content\">\n<p class=\"is-style-warning\" style=\"font-size:0.9em\"><strong>Question:<\/strong> <math data-latex=\"(x - 2)^2 = -6\"><semantics><mrow><mo form=\"prefix\" stretchy=\"false\">(<\/mo><mi>x<\/mi><mo>\u2212<\/mo><mn>2<\/mn><msup><mo form=\"postfix\" stretchy=\"false\">)<\/mo><mn>2<\/mn><\/msup><mo>=<\/mo><mo form=\"prefix\" stretchy=\"false\">\u2212<\/mo><mn>6<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">(x &#8211; 2)^2 = -6<\/annotation><\/semantics><\/math><br>How many distinct real solutions does the given equation have?<br>A) Exactly one<br>B) Exactly two<br>C) Infinitely many<br>D) Zero<\/p>\n\n\n\n<p class=\"is-style-info\" style=\"font-size:0.9em\">\ud83d\udd11 <strong>Key Algebra Rule<\/strong><br>The square of any real number is <strong>never negative<\/strong><br><math display=\"block\"><semantics><mrow><mo stretchy=\"false\">(<\/mo><mtext>real&nbsp;number<\/mtext><msup><mo stretchy=\"false\">)<\/mo><mn>2<\/mn><\/msup><mo>\u2265<\/mo><mn>0<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">(\\text{real number})^2 \\ge 0<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<p class=\"is-style-success\" style=\"font-size:0.9em\">\ud83e\udde0 <strong>Step-by-Step Solution<\/strong><br>Left side:<math display=\"block\"><semantics><mrow><mo stretchy=\"false\">(<\/mo><mi>x<\/mi><mo>\u2212<\/mo><mn>2<\/mn><msup><mo stretchy=\"false\">)<\/mo><mn>2<\/mn><\/msup><mo>\u2265<\/mo><mn>0<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">(x &#8211; 2)^2 \\ge 0<\/annotation><\/semantics><\/math><br>Right side:<math display=\"block\"><semantics><mrow><mo>\u2212<\/mo><mn>6<\/mn><mo>&lt;<\/mo><mn>0<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">-6 &lt; 0<\/annotation><\/semantics><\/math><br>A nonnegative number <strong>cannot equal<\/strong> a negative number.<br>So:<math display=\"block\"><semantics><mrow><mo stretchy=\"false\">(<\/mo><mi>x<\/mi><mo>\u2212<\/mo><mn>2<\/mn><msup><mo stretchy=\"false\">)<\/mo><mn>2<\/mn><\/msup><mo>=<\/mo><mo>\u2212<\/mo><mn>6<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">(x &#8211; 2)^2 = -6<\/annotation><\/semantics><\/math><br>has <strong>no real solution<\/strong>.<br><strong>\u2705 Correct Answer &#8211; Option D : Zero\u200b<\/strong><\/p>\n\n\n\n<p class=\"is-style-error\" style=\"font-size:0.9em\">\u274c <strong>Why Other Options Are Incorrect<\/strong><br><strong>Infinitely many<\/strong> \u274c false<br><strong>Exactly one<\/strong> \u274c impossible<br><strong>Exactly two<\/strong> \u274c no real roots<\/p>\n<\/div><\/details><\/div>\n\n\n\n<div class=\"wp-block-coblocks-accordion-item\"><details><summary class=\"wp-block-coblocks-accordion-item__title\"><strong>17th Question<\/strong><\/summary><div class=\"wp-block-coblocks-accordion-item__content\">\n<p class=\"is-style-warning\" style=\"font-size:0.9em\"><strong>Question:<\/strong><br><math data-latex=\"(ax + 3)(5x^2 - bx + 4) = 20x^3 - 9x^2 - 2x + 12\"><semantics><mrow><mo form=\"prefix\" stretchy=\"false\">(<\/mo><mi>a<\/mi><mi>x<\/mi><mo>+<\/mo><mn>3<\/mn><mo form=\"postfix\" stretchy=\"false\">)<\/mo><mo form=\"prefix\" stretchy=\"false\">(<\/mo><mn>5<\/mn><msup><mi>x<\/mi><mn>2<\/mn><\/msup><mo>\u2212<\/mo><mi>b<\/mi><mi>x<\/mi><mo>+<\/mo><mn>4<\/mn><mo form=\"postfix\" stretchy=\"false\">)<\/mo><mo>=<\/mo><mn>20<\/mn><msup><mi>x<\/mi><mn>3<\/mn><\/msup><mo>\u2212<\/mo><mn>9<\/mn><msup><mi>x<\/mi><mn>2<\/mn><\/msup><mo>\u2212<\/mo><mn>2<\/mn><mi>x<\/mi><mo>+<\/mo><mn>12<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">(ax + 3)(5x^2 &#8211; bx + 4) = 20x^3 &#8211; 9x^2 &#8211; 2x + 12<\/annotation><\/semantics><\/math><br>The equation above is true for all <math data-latex=\"x\"><semantics><mi>x<\/mi><annotation encoding=\"application\/x-tex\">x<\/annotation><\/semantics><\/math>, where <math data-latex=\"a\"><semantics><mi>a<\/mi><annotation encoding=\"application\/x-tex\">a<\/annotation><\/semantics><\/math> and <math data-latex=\"b\"><semantics><mi>b<\/mi><annotation encoding=\"application\/x-tex\">b<\/annotation><\/semantics><\/math> are constants. What is the value of <math data-latex=\"ab\"><semantics><mrow><mi>a<\/mi><mi>b<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">ab<\/annotation><\/semantics><\/math>?<br>A) 18<br>B) 20<br>C) 24<br>D) 40<\/p>\n\n\n\n<p class=\"is-style-info\" style=\"font-size:0.9em\"><strong>1\ufe0f\u20e3 What the Question Is Testing<\/strong><br>This is <strong>polynomial identity matching<\/strong>.<br>When two polynomials are equal <strong>for all x<\/strong>:<br>\ud83d\udc49 <strong>Coefficients of like powers of x must be equal<\/strong><br>This is <strong>not solving for x<\/strong>, this is <strong>comparing coefficients<\/strong>.<\/p>\n\n\n\n<p class=\"is-style-success\" style=\"font-size:0.9em\"><strong>2\ufe0f\u20e3 Step-by-Step Expansion<\/strong><br>Expand the left side carefully.<math display=\"block\"><semantics><mrow><mo stretchy=\"false\">(<\/mo><mi>a<\/mi><mi>x<\/mi><mo>+<\/mo><mn>3<\/mn><mo stretchy=\"false\">)<\/mo><mo stretchy=\"false\">(<\/mo><mn>5<\/mn><msup><mi>x<\/mi><mn>2<\/mn><\/msup><mo>\u2212<\/mo><mi>b<\/mi><mi>x<\/mi><mo>+<\/mo><mn>4<\/mn><mo stretchy=\"false\">)<\/mo><\/mrow><annotation encoding=\"application\/x-tex\">(ax + 3)(5x^2 &#8211; bx + 4)<\/annotation><\/semantics><\/math><br>Distribute term by term:<br>Multiply by <math><semantics><mrow><mi>a<\/mi><mi>x<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">ax<\/annotation><\/semantics><\/math><math display=\"block\"><semantics><mrow><mi>a<\/mi><mi>x<\/mi><mo stretchy=\"false\">(<\/mo><mn>5<\/mn><msup><mi>x<\/mi><mn>2<\/mn><\/msup><mo stretchy=\"false\">)<\/mo><mo>=<\/mo><mn>5<\/mn><mi>a<\/mi><msup><mi>x<\/mi><mn>3<\/mn><\/msup><\/mrow><annotation encoding=\"application\/x-tex\">ax(5x^2) = 5ax^3<\/annotation><\/semantics><\/math><math display=\"block\"><semantics><mrow><mi>a<\/mi><mi>x<\/mi><mo stretchy=\"false\">(<\/mo><mo>\u2212<\/mo><mi>b<\/mi><mi>x<\/mi><mo stretchy=\"false\">)<\/mo><mo>=<\/mo><mo>\u2212<\/mo><mi>a<\/mi><mi>b<\/mi><msup><mi>x<\/mi><mn>2<\/mn><\/msup><\/mrow><annotation encoding=\"application\/x-tex\">ax(-bx) = -abx^2<\/annotation><\/semantics><\/math><math display=\"block\"><semantics><mrow><mi>a<\/mi><mi>x<\/mi><mo stretchy=\"false\">(<\/mo><mn>4<\/mn><mo stretchy=\"false\">)<\/mo><mo>=<\/mo><mn>4<\/mn><mi>a<\/mi><mi>x<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">ax(4) = 4ax<\/annotation><\/semantics><\/math><br>Multiply by <math><semantics><mrow><mn>3<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">3<\/annotation><\/semantics><\/math><math display=\"block\"><semantics><mrow><mn>3<\/mn><mo stretchy=\"false\">(<\/mo><mn>5<\/mn><msup><mi>x<\/mi><mn>2<\/mn><\/msup><mo stretchy=\"false\">)<\/mo><mo>=<\/mo><mn>15<\/mn><msup><mi>x<\/mi><mn>2<\/mn><\/msup><\/mrow><annotation encoding=\"application\/x-tex\">3(5x^2) = 15x^2<\/annotation><\/semantics><\/math><math display=\"block\"><semantics><mrow><mn>3<\/mn><mo stretchy=\"false\">(<\/mo><mo>\u2212<\/mo><mi>b<\/mi><mi>x<\/mi><mo stretchy=\"false\">)<\/mo><mo>=<\/mo><mo>\u2212<\/mo><mn>3<\/mn><mi>b<\/mi><mi>x<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">3(-bx) = -3bx<\/annotation><\/semantics><\/math><math display=\"block\"><semantics><mrow><mn>3<\/mn><mo stretchy=\"false\">(<\/mo><mn>4<\/mn><mo stretchy=\"false\">)<\/mo><mo>=<\/mo><mn>12<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">3(4) = 12<\/annotation><\/semantics><\/math><br>3\ufe0f\u20e3 Combine Like Terms<br><math data-latex=\"5ax^3 - abx^2 + 4ax + 15x^2 - 3bx + 12\\\\ \\\\ 5ax^3 + (-abx^2 + 15x^2) + (4ax -3bx) + 12\"><semantics><mtable columnalign=\"left\" rowspacing=\"0em\"><mtr><mtd style=\"text-align:left\"><mrow><mn>5<\/mn><mi>a<\/mi><msup><mi>x<\/mi><mn>3<\/mn><\/msup><mo>\u2212<\/mo><mi>a<\/mi><mi>b<\/mi><msup><mi>x<\/mi><mn>2<\/mn><\/msup><mo>+<\/mo><mn>4<\/mn><mi>a<\/mi><mi>x<\/mi><mo>+<\/mo><mn>15<\/mn><msup><mi>x<\/mi><mn>2<\/mn><\/msup><mo>\u2212<\/mo><mn>3<\/mn><mi>b<\/mi><mi>x<\/mi><mo>+<\/mo><mn>12<\/mn><\/mrow><mo><\/mo><\/mtd><\/mtr><mtr><mtd style=\"text-align:left\"><mo><\/mo><\/mtd><\/mtr><mtr><mtd style=\"text-align:left\"><mrow><mn>5<\/mn><mi>a<\/mi><msup><mi>x<\/mi><mn>3<\/mn><\/msup><mo>+<\/mo><mo form=\"prefix\" stretchy=\"false\">(<\/mo><mo form=\"prefix\" stretchy=\"false\">\u2212<\/mo><mi>a<\/mi><mi>b<\/mi><msup><mi>x<\/mi><mn>2<\/mn><\/msup><mo>+<\/mo><mn>15<\/mn><msup><mi>x<\/mi><mn>2<\/mn><\/msup><mo form=\"postfix\" stretchy=\"false\">)<\/mo><mo>+<\/mo><mo form=\"prefix\" stretchy=\"false\">(<\/mo><mn>4<\/mn><mi>a<\/mi><mi>x<\/mi><mo>\u2212<\/mo><mn>3<\/mn><mi>b<\/mi><mi>x<\/mi><mo form=\"postfix\" stretchy=\"false\">)<\/mo><mo>+<\/mo><mn>12<\/mn><\/mrow><\/mtd><\/mtr><\/mtable><annotation encoding=\"application\/x-tex\">5ax^3 &#8211; abx^2 + 4ax + 15x^2 &#8211; 3bx + 12\\\\ \\\\ 5ax^3 + (-abx^2 + 15x^2) + (4ax -3bx) + 12<\/annotation><\/semantics><\/math><br><br>4\ufe0f\u20e3 Match with the Right Side<br>Right side:<math display=\"block\"><semantics><mrow><mn>20<\/mn><msup><mi>x<\/mi><mn>3<\/mn><\/msup><mo>\u2212<\/mo><mn>9<\/mn><msup><mi>x<\/mi><mn>2<\/mn><\/msup><mo>\u2212<\/mo><mn>2<\/mn><mi>x<\/mi><mo>+<\/mo><mn>12<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">20x^3 &#8211; 9x^2 &#8211; 2x + 12<\/annotation><\/semantics><\/math><br>Now match <strong>power by power<\/strong>:<br><math data-latex=\"5ax^3 + (-abx^2 + 15x^2) + (4ax -3bx) + 12\\ =\\ 20x^3 - 9x^2 - 2x + 12\"><semantics><mrow><mn>5<\/mn><mi>a<\/mi><msup><mi>x<\/mi><mn>3<\/mn><\/msup><mo>+<\/mo><mo form=\"prefix\" stretchy=\"false\">(<\/mo><mo form=\"prefix\" stretchy=\"false\">\u2212<\/mo><mi>a<\/mi><mi>b<\/mi><msup><mi>x<\/mi><mn>2<\/mn><\/msup><mo>+<\/mo><mn>15<\/mn><msup><mi>x<\/mi><mn>2<\/mn><\/msup><mo form=\"postfix\" stretchy=\"false\">)<\/mo><mo>+<\/mo><mo form=\"prefix\" stretchy=\"false\">(<\/mo><mn>4<\/mn><mi>a<\/mi><mi>x<\/mi><mo>\u2212<\/mo><mn>3<\/mn><mi>b<\/mi><mi>x<\/mi><mo form=\"postfix\" stretchy=\"false\">)<\/mo><mo>+<\/mo><mn>12<\/mn><mtext>&nbsp;<\/mtext><mo>=<\/mo><mtext>&nbsp;<\/mtext><mn>20<\/mn><msup><mi>x<\/mi><mn>3<\/mn><\/msup><mo>\u2212<\/mo><mn>9<\/mn><msup><mi>x<\/mi><mn>2<\/mn><\/msup><mo>\u2212<\/mo><mn>2<\/mn><mi>x<\/mi><mo>+<\/mo><mn>12<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">5ax^3 + (-abx^2 + 15x^2) + (4ax -3bx) + 12\\ =\\ 20x^3 &#8211; 9x^2 &#8211; 2x + 12<\/annotation><\/semantics><\/math><br>Now solve it using matching values \/ figures.<br><br><math><semantics><mrow><msup><mi>x<\/mi><mn>3<\/mn><\/msup><\/mrow><annotation encoding=\"application\/x-tex\">x^3<\/annotation><\/semantics><\/math> terms: <math data-latex=\"5ax^3 = 20x^3\"><semantics><mrow><mn>5<\/mn><mi>a<\/mi><msup><mi>x<\/mi><mn>3<\/mn><\/msup><mo>=<\/mo><mn>20<\/mn><msup><mi>x<\/mi><mn>3<\/mn><\/msup><\/mrow><annotation encoding=\"application\/x-tex\">5ax^3 = 20x^3<\/annotation><\/semantics><\/math><br>Remove <math data-latex=\"x^3\"><semantics><msup><mi>x<\/mi><mn>3<\/mn><\/msup><annotation encoding=\"application\/x-tex\">x^3<\/annotation><\/semantics><\/math> from both sides: <math data-latex=\"5a = 20\"><semantics><mrow><mn>5<\/mn><mi>a<\/mi><mo>=<\/mo><mn>20<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">5a = 20<\/annotation><\/semantics><\/math><br><math data-latex=\"\\\\a = \\frac{20}{5}\\\\ \\\\a = 4\"><semantics><mtable columnalign=\"left\" rowspacing=\"0em\"><mtr><mtd style=\"text-align:left\"><mo><\/mo><\/mtd><\/mtr><mtr><mtd style=\"text-align:left\"><mrow><mi>a<\/mi><mo>=<\/mo><mfrac><mn>20<\/mn><mn>5<\/mn><\/mfrac><\/mrow><mo><\/mo><\/mtd><\/mtr><mtr><mtd style=\"text-align:left\"><mo><\/mo><\/mtd><\/mtr><mtr><mtd style=\"text-align:left\"><mrow><mi>a<\/mi><mo>=<\/mo><mn>4<\/mn><\/mrow><\/mtd><\/mtr><\/mtable><annotation encoding=\"application\/x-tex\">\\\\a = \\frac{20}{5}\\\\ \\\\a = 4<\/annotation><\/semantics><\/math><br><br><math><semantics><mrow><msup><mi>x<\/mi><mn>2<\/mn><\/msup><\/mrow><annotation encoding=\"application\/x-tex\">x^2<\/annotation><\/semantics><\/math> terms: <math data-latex=\"(-abx^2 + 15x^2) = -9x^2\"><semantics><mrow><mo form=\"prefix\" stretchy=\"false\">(<\/mo><mo form=\"prefix\" stretchy=\"false\">\u2212<\/mo><mi>a<\/mi><mi>b<\/mi><msup><mi>x<\/mi><mn>2<\/mn><\/msup><mo>+<\/mo><mn>15<\/mn><msup><mi>x<\/mi><mn>2<\/mn><\/msup><mo form=\"postfix\" stretchy=\"false\">)<\/mo><mo>=<\/mo><mo form=\"prefix\" stretchy=\"false\">\u2212<\/mo><mn>9<\/mn><msup><mi>x<\/mi><mn>2<\/mn><\/msup><\/mrow><annotation encoding=\"application\/x-tex\">(-abx^2 + 15x^2) = -9x^2<\/annotation><\/semantics><\/math><br>Remove <math data-latex=\"x^2\"><semantics><msup><mi>x<\/mi><mn>2<\/mn><\/msup><annotation encoding=\"application\/x-tex\">x^2<\/annotation><\/semantics><\/math> from both sides: <math data-latex=\"-ab + 15 = -9\"><semantics><mrow><mo>\u2212<\/mo><mi>a<\/mi><mi>b<\/mi><mo>+<\/mo><mn>15<\/mn><mo>=<\/mo><mo form=\"prefix\" stretchy=\"false\">\u2212<\/mo><mn>9<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">-ab + 15 = -9<\/annotation><\/semantics><\/math><br><math data-latex=\"-ab = -15 - 9\\\\ \\\\ -ab = -24\"><semantics><mtable columnalign=\"left\" rowspacing=\"0em\"><mtr><mtd style=\"text-align:left\"><mrow><mo>\u2212<\/mo><mi>a<\/mi><mi>b<\/mi><mo>=<\/mo><mo form=\"prefix\" stretchy=\"false\">\u2212<\/mo><mn>15<\/mn><mo>\u2212<\/mo><mn>9<\/mn><\/mrow><mo><\/mo><\/mtd><\/mtr><mtr><mtd style=\"text-align:left\"><mo><\/mo><\/mtd><\/mtr><mtr><mtd style=\"text-align:left\"><mrow><mo>\u2212<\/mo><mi>a<\/mi><mi>b<\/mi><mo>=<\/mo><mo form=\"prefix\" stretchy=\"false\">\u2212<\/mo><mn>24<\/mn><\/mrow><\/mtd><\/mtr><\/mtable><annotation encoding=\"application\/x-tex\">-ab = -15 &#8211; 9\\\\ \\\\ -ab = -24<\/annotation><\/semantics><\/math><br>Remove negative from both sides: <math data-latex=\"ab = 24\"><semantics><mrow><mi>a<\/mi><mi>b<\/mi><mo>=<\/mo><mn>24<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">ab = 24<\/annotation><\/semantics><\/math><br><br>We already have the value. We know <strong>ab<\/strong>, so we do not need to calculate anymore, but if it is something else, this is how you proceed.<br><br>\u2705 <strong>Correct Answer: Option C<\/strong><br><math data-latex=\"ab = 24\"><semantics><mrow><mi>a<\/mi><mi>b<\/mi><mo>=<\/mo><mn>24<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">ab = 24<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<p class=\"is-style-error\" style=\"font-size:0.9em\">\u2022 <strong>18 \/ 20<\/strong> \u2192 students forget to include <strong>+15<\/strong> from distribution<br>\u2022 <strong>40<\/strong> \u2192 students match only <math><semantics><mrow><mn>5<\/mn><mi>a<\/mi><mo>=<\/mo><mn>20<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">5a = 20<\/annotation><\/semantics><\/math> and multiply randomly<br>\u2022 SAT loves coefficient-matching traps<\/p>\n\n\n\n<p class=\"is-style-info\" style=\"font-size:0.9em\"><strong><em>Use Desmos as a calculator only for this one.<\/em><\/strong><\/p>\n<\/div><\/details><\/div>\n\n\n\n<div class=\"wp-block-coblocks-accordion-item\"><details><summary class=\"wp-block-coblocks-accordion-item__title\"><strong>18th Question<\/strong><\/summary><div class=\"wp-block-coblocks-accordion-item__content\">\n<p class=\"is-style-warning\" style=\"font-size:0.9em\"><strong>Question:<\/strong> A manufacturing plant makes 10-inch, 9-inch, and 7-inch frying pans. During a certain day, the number of 10-inch frying pans that the manufacturing plant makes is 4 times the number n of 9-inch frying pans it makes, and the number of 7-inch frying pans it makes is 10. During this day, the manufacturing plant makes 100 frying pans total. Which equation represents this situation?<br>A) <math data-latex=\"10(4n) + 9n + 7(10) = 100\"><semantics><mrow><mn>10<\/mn><mo form=\"prefix\" stretchy=\"false\">(<\/mo><mn>4<\/mn><mi>n<\/mi><mo form=\"postfix\" stretchy=\"false\">)<\/mo><mo>+<\/mo><mn>9<\/mn><mi>n<\/mi><mo>+<\/mo><mn>7<\/mn><mo form=\"prefix\" stretchy=\"false\">(<\/mo><mn>10<\/mn><mo form=\"postfix\" stretchy=\"false\">)<\/mo><mo>=<\/mo><mn>100<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">10(4n) + 9n + 7(10) = 100<\/annotation><\/semantics><\/math><br>B) <math data-latex=\"10n + 9n + 7n = 100\"><semantics><mrow><mn>10<\/mn><mi>n<\/mi><mo>+<\/mo><mn>9<\/mn><mi>n<\/mi><mo>+<\/mo><mn>7<\/mn><mi>n<\/mi><mo>=<\/mo><mn>100<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">10n + 9n + 7n = 100<\/annotation><\/semantics><\/math><br>C) <math data-latex=\"4n + 10 = 100\"><semantics><mrow><mn>4<\/mn><mi>n<\/mi><mo>+<\/mo><mn>10<\/mn><mo>=<\/mo><mn>100<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">4n + 10 = 100<\/annotation><\/semantics><\/math><br>D) <math data-latex=\"5n + 10 = 100\"><semantics><mrow><mn>5<\/mn><mi>n<\/mi><mo>+<\/mo><mn>10<\/mn><mo>=<\/mo><mn>100<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">5n + 10 = 100<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<p class=\"is-style-info\" style=\"font-size:0.9em\"><strong>Understand the Question<\/strong><br><strong>Problem (understanding the situation)<\/strong><br>A manufacturing plant makes <strong>three sizes of frying pans<\/strong>:<br>10-inch pans<br>9-inch pans<br>7-inch pans<br><br>Given:<br>Number of <strong>9-inch pans = n<\/strong><br>Number of <strong>10-inch pans = 4n<\/strong><br>Number of <strong>7-inch pans = 10<\/strong><br><strong>Total pans made = 100<\/strong><br>The question asks:<br>\ud83d\udc49 <strong>Which equation correctly represents this situation?<\/strong><br><br><strong>Important Concept \/ Rule<\/strong><br>This is a <strong>word-to-equation translation<\/strong> problem.<br>Key idea: When counting objects, <strong>we add the number of objects<\/strong>, not their sizes.<br>\u26a0\ufe0f The inches (10-inch, 9-inch, 7-inch) are <strong>labels<\/strong>, NOT values to multiply.<\/p>\n\n\n\n<p class=\"is-style-success\" style=\"font-size:0.9em\"><strong>Step-by-Step Construction of the Equation<\/strong><br>10-inch pans \u2192 <strong>4n pans<\/strong><br>9-inch pans \u2192 <strong>n pans<\/strong><br>7-inch pans \u2192 <strong>10 pans<\/strong><br>Total pans:<math display=\"block\"><semantics><mrow><mn>4<\/mn><mi>n<\/mi><mo>+<\/mo><mi>n<\/mi><mo>+<\/mo><mn>10<\/mn><mo>=<\/mo><mn>100<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">4n + n + 10 = 100<\/annotation><\/semantics><\/math><br>Simplify:<math display=\"block\"><semantics><mrow><mn>5<\/mn><mi>n<\/mi><mo>+<\/mo><mn>10<\/mn><mo>=<\/mo><mn>100<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">5n + 10 = 100<\/annotation><\/semantics><\/math><br><strong>Correct Answer:<\/strong><math display=\"block\"><semantics><mrow><menclose notation=\"box\"><mstyle scriptlevel=\"0\" displaystyle=\"false\"><mstyle scriptlevel=\"0\" displaystyle=\"false\"><mstyle scriptlevel=\"0\" displaystyle=\"true\"><mrow><mn>5<\/mn><mi>n<\/mi><mo>+<\/mo><mn>10<\/mn><mo>=<\/mo><mn>100<\/mn><\/mrow><\/mstyle><\/mstyle><\/mstyle><\/menclose><\/mrow><annotation encoding=\"application\/x-tex\">\\boxed{5n + 10 = 100}<\/annotation><\/semantics><\/math>\u2705 <strong>Option D<\/strong><\/p>\n\n\n\n<p class=\"is-style-error\" style=\"font-size:0.9em\"><strong>Why Other Options Are Incorrect<\/strong><br><strong>Option A:<\/strong><math display=\"block\"><semantics><mrow><mn>10<\/mn><mo stretchy=\"false\">(<\/mo><mn>4<\/mn><mi>n<\/mi><mo stretchy=\"false\">)<\/mo><mo>+<\/mo><mn>9<\/mn><mi>n<\/mi><mo>+<\/mo><mn>7<\/mn><mo stretchy=\"false\">(<\/mo><mn>10<\/mn><mo stretchy=\"false\">)<\/mo><\/mrow><annotation encoding=\"application\/x-tex\">10(4n) + 9n + 7(10)<\/annotation><\/semantics><\/math><br>\u274c Incorrectly multiplies <strong>pan size<\/strong> with quantity<br>Pan size does NOT affect the count<br><br><strong>Option B:<\/strong><math display=\"block\"><semantics><mrow><mn>10<\/mn><mi>n<\/mi><mo>+<\/mo><mn>9<\/mn><mi>n<\/mi><mo>+<\/mo><mn>7<\/mn><mi>n<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">10n + 9n + 7n<\/annotation><\/semantics><\/math><br>\u274c Assumes all quantities are proportional to size \u2014 false<br><br><strong>Option C:<\/strong><math display=\"block\"><semantics><mrow><mn>4<\/mn><mi>n<\/mi><mo>+<\/mo><mn>10<\/mn><mo>=<\/mo><mn>100<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">4n + 10 = 100<\/annotation><\/semantics><\/math><br>\u274c Ignores the <strong>n 9-inch pans<\/strong><br><br><strong>Common Student Mistakes<\/strong><br>\u274c Multiplying by pan sizes<br>\u274c Forgetting one category<br>\u274c Treating inches as quantities<\/p>\n<\/div><\/details><\/div>\n\n\n\n<div class=\"wp-block-coblocks-accordion-item\"><details><summary class=\"wp-block-coblocks-accordion-item__title\"><strong>19th Question<\/strong><\/summary><div class=\"wp-block-coblocks-accordion-item__content\">\n<p class=\"is-style-warning\" style=\"font-size:0.9em\"><strong>Question:<\/strong> Near the end of a US cable news show, the host invited viewers to respond to a poll on the show\u2019s website that asked, \u201cDo you support the new federal policy discussed during the show?\u201d At the end of the show, the host reported that 28% responded \u201cYes,\u201d and 70% responded \u201cNo.\u201d Which of the following best explains why the results are unlikely to represent the sentiments of the population of the United States?<br>A) The percentages do not add up to 100%, so any possible conclusions from the poll are invalid.<br>B) Those who responded to the poll were not a random sample of the population of the United States.<br>C) There were not 50% \u201cYes\u201d responses and 50% \u201cNo\u201d responses.<br>D) The show did not allow viewers enough time to respond to the poll.<\/p>\n\n\n\n<p class=\"is-style-info\" style=\"font-size:0.9em\">\ud83d\udd11 <strong>Key Statistical Concept<\/strong><br>This is a <strong>sampling bias<\/strong> question.<br>A <strong>valid survey<\/strong> requires:<br>~ Random selection<br>~ Equal chance for all members of the population to respond<\/p>\n\n\n\n<p class=\"is-style-success\" style=\"font-size:0.9em\">\ud83e\udde0 <strong>Step-by-Step Explanation<\/strong><br>The poll was taken from <strong>viewers of one cable news show<\/strong><br>Only people who:<br>~ Watch that show<br>~ Visit the website<br>~ Choose to respond<br>were included<br>This means the sample:<br>~ Is <strong>self-selected<\/strong><br>~ Is <strong>not random<\/strong><br>~ Likely reflects opinions of a <strong>specific audience<\/strong>, not the US population<br><br>\u2705 <strong>Correct Answer<\/strong><br>B) Those who responded to the poll were not a random sample of the population of the United States.\u200b<\/p>\n\n\n\n<p class=\"is-style-error\" style=\"font-size:0.9em\">\u274c <strong>Why Other Options Are Incorrect<\/strong><br><strong>Option A<\/strong><br>\u274c Percentages not totaling 100% could be rounding \u2014 not the main issue.<br><br><strong>Option C<\/strong><br>\u274c Surveys do <strong>not<\/strong> need 50\u201350 results to be valid.<br><br><strong>Option D<\/strong><br>\u274c Time allowed does not fix <strong>sampling bias<\/strong>.<\/p>\n<\/div><\/details><\/div>\n\n\n\n<div class=\"wp-block-coblocks-accordion-item\"><details><summary class=\"wp-block-coblocks-accordion-item__title\"><strong>20th Question<\/strong><\/summary><div class=\"wp-block-coblocks-accordion-item__content\">\n<p class=\"is-style-warning\" style=\"font-size:0.9em\"><strong>Question:<\/strong><br><math data-latex=\"ax + by = 72 \\\\ 6x + 2by = 56\"><semantics><mtable columnalign=\"left\" rowspacing=\"0em\"><mtr><mtd style=\"text-align:left\"><mrow><mi>a<\/mi><mi>x<\/mi><mo>+<\/mo><mi>b<\/mi><mi>y<\/mi><mo>=<\/mo><mn>72<\/mn><\/mrow><mo><\/mo><\/mtd><\/mtr><mtr><mtd style=\"text-align:left\"><mrow><mn>6<\/mn><mi>x<\/mi><mo>+<\/mo><mn>2<\/mn><mi>b<\/mi><mi>y<\/mi><mo>=<\/mo><mn>56<\/mn><\/mrow><\/mtd><\/mtr><\/mtable><annotation encoding=\"application\/x-tex\">ax + by = 72 \\\\ 6x + 2by = 56<\/annotation><\/semantics><\/math><br>In the given system of equations, <math data-latex=\"a\"><semantics><mi>a<\/mi><annotation encoding=\"application\/x-tex\">a<\/annotation><\/semantics><\/math> and <math data-latex=\"b\"><semantics><mi>b<\/mi><annotation encoding=\"application\/x-tex\">b<\/annotation><\/semantics><\/math> are constants. The graphs of these equations in the <em>xy<\/em>-plane intersect at the point (4, <em>y<\/em>). What is the value of <math data-latex=\"a\"><semantics><mi>a<\/mi><annotation encoding=\"application\/x-tex\">a<\/annotation><\/semantics><\/math>?<br>A) 14<br>B) 6<br>C) 4<br>D) 3<\/p>\n\n\n\n<p class=\"is-style-success\" style=\"font-size:0.9em\"><strong>Choice A<\/strong> is correct.<br><strong>\ud83e\uddee Step-by-Step Solution<\/strong><br>~ <em>xy<\/em>-plane (<em>x, y<\/em>) = (4, <em>y<\/em>)<br>~ Because the point of intersection is <math><semantics><mrow><mo stretchy=\"false\">(<\/mo><mn>4<\/mn><mo separator=\"true\">,<\/mo><mi>y<\/mi><mo stretchy=\"false\">)<\/mo><\/mrow><annotation encoding=\"application\/x-tex\">(4, y)<\/annotation><\/semantics><\/math>,<br>~ <strong>both equations must be true when <em>x<\/em> = 4<\/strong>.<br>That is the key instruction hidden in the question.<br><br><math data-latex=\"ax + by = 72 \\\\ 6x + 2by = 56\"><semantics><mtable columnalign=\"left\" rowspacing=\"0em\"><mtr><mtd style=\"text-align:left\"><mrow><mi>a<\/mi><mi>x<\/mi><mo>+<\/mo><mi>b<\/mi><mi>y<\/mi><mo>=<\/mo><mn>72<\/mn><\/mrow><mo><\/mo><\/mtd><\/mtr><mtr><mtd style=\"text-align:left\"><mrow><mn>6<\/mn><mi>x<\/mi><mo>+<\/mo><mn>2<\/mn><mi>b<\/mi><mi>y<\/mi><mo>=<\/mo><mn>56<\/mn><\/mrow><\/mtd><\/mtr><\/mtable><annotation encoding=\"application\/x-tex\">ax + by = 72 \\\\ 6x + 2by = 56<\/annotation><\/semantics><\/math> <br><strong>Step 1: Substitute <\/strong><math><semantics><mrow><mi>x<\/mi><mo>=<\/mo><mn>4<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">x = 4<\/annotation><\/semantics><\/math><strong> into the first equation<\/strong><br><math display=\"block\"><semantics><mrow><mi>a<\/mi><mo stretchy=\"false\">(<\/mo><mn>4<\/mn><mo stretchy=\"false\">)<\/mo><mo>+<\/mo><mi>b<\/mi><mi>y<\/mi><mo>=<\/mo><mn>72<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">a(4) + by = 72<\/annotation><\/semantics><\/math><br><math display=\"block\"><semantics><mrow><mn>4<\/mn><mi>a<\/mi><mo>+<\/mo><mi>b<\/mi><mi>y<\/mi><mo>=<\/mo><mn>72<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">4a + by = 72<\/annotation><\/semantics><\/math><br>Rearrange:<br><math display=\"block\"><semantics><mrow><mi>b<\/mi><mi>y<\/mi><mo>=<\/mo><mn>72<\/mn><mo>\u2212<\/mo><mn>4<\/mn><mi>a<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">by = 72 &#8211; 4a<\/annotation><\/semantics><\/math><br><strong>Step 2: Substitute <\/strong><math><semantics><mrow><mi>x<\/mi><mo>=<\/mo><mn>4<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">x = 4<\/annotation><\/semantics><\/math><strong> into the second equation<\/strong><br><math display=\"block\"><semantics><mrow><mn>6<\/mn><mo stretchy=\"false\">(<\/mo><mn>4<\/mn><mo stretchy=\"false\">)<\/mo><mo>+<\/mo><mn>2<\/mn><mi>b<\/mi><mi>y<\/mi><mo>=<\/mo><mn>56<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">6(4) + 2by = 56<\/annotation><\/semantics><\/math><br><math display=\"block\"><semantics><mrow><mn>24<\/mn><mo>+<\/mo><mn>2<\/mn><mi>b<\/mi><mi>y<\/mi><mo>=<\/mo><mn>56<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">24 + 2by = 56<\/annotation><\/semantics><\/math><br>Subtract 24:<br><math display=\"block\"><semantics><mrow><mn>2<\/mn><mi>b<\/mi><mi>y<\/mi><mo>=<\/mo><mn>32<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">2by = 32<\/annotation><\/semantics><\/math><br>Divide by 2:<br><math display=\"block\"><semantics><mrow><mi>b<\/mi><mi>y<\/mi><mo>=<\/mo><mn>16<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">by = 16<\/annotation><\/semantics><\/math><br><strong>Step 3: Equate the two expressions for <\/strong><math><semantics><mrow><mi>b<\/mi><mi>y<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">by<\/annotation><\/semantics><\/math><br>From above:<br><math display=\"block\"><semantics><mrow><mn>72<\/mn><mo>\u2212<\/mo><mn>4<\/mn><mi>a<\/mi><mo>=<\/mo><mn>16<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">72 &#8211; 4a = 16<\/annotation><\/semantics><\/math><br>Solve:<br><math display=\"block\"><semantics><mrow><mn>4<\/mn><mi>a<\/mi><mo>=<\/mo><mn>56<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">4a = 56<\/annotation><\/semantics><\/math><br><math display=\"block\"><semantics><mrow><mi>a<\/mi><mo>=<\/mo><mn>14<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">a = 14<\/annotation><\/semantics><\/math><br>\u2705 Correct Answer: <strong>A) 14<\/strong><\/p>\n\n\n\n<p class=\"is-style-error\" style=\"font-size:0.9em\"><strong>Option B: 6 \u274c<\/strong><br><strong>Trap:<\/strong> Student divides 56 by 4 directly, ignoring system structure.<\/p>\n\n\n\n<p class=\"is-style-error\" style=\"font-size:0.9em\"><strong>Option C: 4 \u274c<\/strong><br><strong>Trap:<\/strong> Student confuses <math><semantics><mrow><mi>x<\/mi><mo>=<\/mo><mn>4<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">x = 4<\/annotation><\/semantics><\/math> with <math><semantics><mrow><mi>a<\/mi><mo>=<\/mo><mn>4<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">a = 4<\/annotation><\/semantics><\/math>.<\/p>\n\n\n\n<p class=\"is-style-error\" style=\"font-size:0.9em\"><strong>Option D: 3 \u274c<\/strong><br><strong>Trap:<\/strong> Student divides 12 by 4 after incorrect simplification.<\/p>\n\n\n\n<p class=\"is-style-info\" style=\"font-size:0.9em\"><strong>\ud83e\uddee DESMOS CONFIRMATION<\/strong><br>1. Open <strong>Desmos<\/strong><br>2. Enter: <br>ax + by = 72<br>6x + 2by = 56<br>3. Substitute <math><semantics><mrow><mi>x<\/mi><mo>=<\/mo><mn>4<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">x = 4<\/annotation><\/semantics><\/math> and test values of <math><semantics><mrow><mi>a<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">a<\/annotation><\/semantics><\/math><br>4. Only <strong>a = 14<\/strong> gives a common intersection point<br>\u2714 Confirmed<\/p>\n<\/div><\/details><\/div>\n\n\n\n<div class=\"wp-block-coblocks-accordion-item\"><details><summary class=\"wp-block-coblocks-accordion-item__title\"><strong>21th Question<\/strong><\/summary><div class=\"wp-block-coblocks-accordion-item__content\">\n<p class=\"is-style-warning\" style=\"font-size:0.9em\"><strong>Question:<\/strong> The product of two positive integers is 462. If the first integer is 5 greater than twice the second integer, what is the smaller of the two integers?<br><br>[Type-Based Answer: In the final exam, you will type the answer rather than choose from options.]<\/p>\n\n\n\n<p class=\"is-style-info\"><strong>\ud83d\udd39 Understand the QUESTION<br>Given<\/strong><br>~ The product of two <strong>positive integers<\/strong> is <strong>462<\/strong><br>~ The <strong>first integer<\/strong> is <strong>5 greater than twice the second integer<\/strong><br>~ Find the <strong>smaller integer<\/strong><\/p>\n\n\n\n<p class=\"is-style-success\" style=\"font-size:0.9em\"><strong>\ud83e\udde0 STEP 1: DEFINE VARIABLES (WHY THIS SETUP)<\/strong><br>There are two integers.<br>Let the second integer: The smaller integer = <math><semantics><mrow><mi>x<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">x<\/annotation><\/semantics><\/math><br>Then the first integer is: Twice the second integer = <math><semantics><mrow><mn>2<\/mn><mi>x<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">2x<\/annotation><\/semantics><\/math> with 5 greater<br>~ First integer = <math><semantics><mrow><mn>2<\/mn><mi>x<\/mi><mo>+<\/mo><mn>5<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">2x + 5<\/annotation><\/semantics><\/math><br>This setup is chosen because:<br>~ SAT word problems <strong>always describe one quantity in terms of another<\/strong><br>~ \u201c5 greater than twice\u201d means <strong>multiply first, then add<\/strong><br><br><strong>\ud83e\uddee STEP 2: USE THE PRODUCT CONDITION<\/strong><br>The product is 462, so: <strong><em>2nd Integer(1st Integer) = Total<\/em><\/strong><br><math display=\"block\"><semantics><mrow><mi>x<\/mi><mo stretchy=\"false\">(<\/mo><mn>2<\/mn><mi>x<\/mi><mo>+<\/mo><mn>5<\/mn><mo stretchy=\"false\">)<\/mo><mo>=<\/mo><mn>462<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">x(2x + 5) = 462<\/annotation><\/semantics><\/math><br>Expand:<math display=\"block\"><semantics><mrow><mn>2<\/mn><msup><mi>x<\/mi><mn>2<\/mn><\/msup><mo>+<\/mo><mn>5<\/mn><mi>x<\/mi><mo>=<\/mo><mn>462<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">2x^2 + 5x = 462<\/annotation><\/semantics><\/math><br>Move everything to one side:<math display=\"block\"><semantics><mrow><mn>2<\/mn><msup><mi>x<\/mi><mn>2<\/mn><\/msup><mo>+<\/mo><mn>5<\/mn><mi>x<\/mi><mo>\u2212<\/mo><mn>462<\/mn><mo>=<\/mo><mn>0<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">2x^2 + 5x &#8211; 462 = 0<\/annotation><\/semantics><\/math><br>This step is necessary because:<br>~ Quadratic equations must be written in <strong>standard form<\/strong> to factor or solve<br><strong>Method 1:<br><\/strong><math data-latex=\"\\\\ 2x^2 + 5x - 462 = 0\"><semantics><mtable columnalign=\"left\" rowspacing=\"0em\"><mtr><mtd style=\"text-align:left\"><mo><\/mo><\/mtd><\/mtr><mtr><mtd style=\"text-align:left\"><mrow><mn>2<\/mn><msup><mi>x<\/mi><mn>2<\/mn><\/msup><mo>+<\/mo><mn>5<\/mn><mi>x<\/mi><mo>\u2212<\/mo><mn>462<\/mn><mo>=<\/mo><mn>0<\/mn><\/mrow><\/mtd><\/mtr><\/mtable><annotation encoding=\"application\/x-tex\">\\\\ 2x^2 + 5x &#8211; 462 = 0<\/annotation><\/semantics><\/math><br>For quadratic, we need to prepare these first: <math data-latex=\"ax^2 + bx + c = 0\"><semantics><mrow><mi>a<\/mi><msup><mi>x<\/mi><mn>2<\/mn><\/msup><mo>+<\/mo><mi>b<\/mi><mi>x<\/mi><mo>+<\/mo><mi>c<\/mi><mo>=<\/mo><mn>0<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">ax^2 + bx + c = 0<\/annotation><\/semantics><\/math><br> we will turn <strong>b<\/strong> into this <math data-latex=\"ax^2 + (b_1 + b_2)x + c = 0 \"><semantics><mrow><mi>a<\/mi><msup><mi>x<\/mi><mn>2<\/mn><\/msup><mo>+<\/mo><mo form=\"prefix\" stretchy=\"false\">(<\/mo><msub><mi>b<\/mi><mn>1<\/mn><\/msub><mo>+<\/mo><msub><mi>b<\/mi><mn>2<\/mn><\/msub><mo form=\"postfix\" stretchy=\"false\">)<\/mo><mi>x<\/mi><mo>+<\/mo><mi>c<\/mi><mo>=<\/mo><mn>0<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">ax^2 + (b_1 + b_2)x + c = 0 <\/annotation><\/semantics><\/math> then to <math data-latex=\"(ax^2 + b_1) + (b_2 + c) = 0\"><semantics><mrow><mo form=\"prefix\" stretchy=\"false\">(<\/mo><mi>a<\/mi><msup><mi>x<\/mi><mn>2<\/mn><\/msup><mo>+<\/mo><msub><mi>b<\/mi><mn>1<\/mn><\/msub><mo form=\"postfix\" stretchy=\"false\">)<\/mo><mo>+<\/mo><mo form=\"prefix\" stretchy=\"false\">(<\/mo><msub><mi>b<\/mi><mn>2<\/mn><\/msub><mo>+<\/mo><mi>c<\/mi><mo form=\"postfix\" stretchy=\"false\">)<\/mo><mo>=<\/mo><mn>0<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">(ax^2 + b_1) + (b_2 + c) = 0<\/annotation><\/semantics><\/math><br>After that just simplify it:<br><math data-latex=\"\\\\ ax = 2x \\\\ \\\\ax^2 = 2x^2\\\\ \\\\bx = 5x\"><semantics><mtable columnalign=\"left\" rowspacing=\"0em\"><mtr><mtd style=\"text-align:left\"><mo><\/mo><\/mtd><\/mtr><mtr><mtd style=\"text-align:left\"><mrow><mi>a<\/mi><mi>x<\/mi><mo>=<\/mo><mn>2<\/mn><mi>x<\/mi><\/mrow><mo><\/mo><\/mtd><\/mtr><mtr><mtd style=\"text-align:left\"><mo><\/mo><\/mtd><\/mtr><mtr><mtd style=\"text-align:left\"><mrow><mi>a<\/mi><msup><mi>x<\/mi><mn>2<\/mn><\/msup><mo>=<\/mo><mn>2<\/mn><msup><mi>x<\/mi><mn>2<\/mn><\/msup><\/mrow><mo><\/mo><\/mtd><\/mtr><mtr><mtd style=\"text-align:left\"><mo><\/mo><\/mtd><\/mtr><mtr><mtd style=\"text-align:left\"><mrow><mi>b<\/mi><mi>x<\/mi><mo>=<\/mo><mn>5<\/mn><mi>x<\/mi><\/mrow><\/mtd><\/mtr><\/mtable><annotation encoding=\"application\/x-tex\">\\\\ ax = 2x \\\\ \\\\ax^2 = 2x^2\\\\ \\\\bx = 5x<\/annotation><\/semantics><\/math><br><math data-latex=\"b = 5 = (b_1 + b_2)\"><semantics><mrow><mi>b<\/mi><mo>=<\/mo><mn>5<\/mn><mo>=<\/mo><mo form=\"prefix\" stretchy=\"false\">(<\/mo><msub><mi>b<\/mi><mn>1<\/mn><\/msub><mo>+<\/mo><msub><mi>b<\/mi><mn>2<\/mn><\/msub><mo form=\"postfix\" stretchy=\"false\">)<\/mo><\/mrow><annotation encoding=\"application\/x-tex\">b = 5 = (b_1 + b_2)<\/annotation><\/semantics><\/math> , so the best values suit here are (-28 + 33) = 5<br><math data-latex=\"\\\\ b_1 = -28\\\\ \\\\ b_2 = 33\\\\ \\\\c = -462\"><semantics><mtable columnalign=\"left\" rowspacing=\"0em\"><mtr><mtd style=\"text-align:left\"><mo><\/mo><\/mtd><\/mtr><mtr><mtd style=\"text-align:left\"><mrow><msub><mi>b<\/mi><mn>1<\/mn><\/msub><mo>=<\/mo><mo form=\"prefix\" stretchy=\"false\">\u2212<\/mo><mn>28<\/mn><\/mrow><mo><\/mo><\/mtd><\/mtr><mtr><mtd style=\"text-align:left\"><mo><\/mo><\/mtd><\/mtr><mtr><mtd style=\"text-align:left\"><mrow><msub><mi>b<\/mi><mn>2<\/mn><\/msub><mo>=<\/mo><mn>33<\/mn><\/mrow><mo><\/mo><\/mtd><\/mtr><mtr><mtd style=\"text-align:left\"><mo><\/mo><\/mtd><\/mtr><mtr><mtd style=\"text-align:left\"><mrow><mi>c<\/mi><mo>=<\/mo><mo form=\"prefix\" stretchy=\"false\">\u2212<\/mo><mn>462<\/mn><\/mrow><\/mtd><\/mtr><\/mtable><annotation encoding=\"application\/x-tex\">\\\\ b_1 = -28\\\\ \\\\ b_2 = 33\\\\ \\\\c = -462<\/annotation><\/semantics><\/math><br><br>Here is a twist the value c made out of <strong>2 integer<\/strong>, so <math data-latex=\"b_1 \\times b_2\"><semantics><mrow><msub><mi>b<\/mi><mn>1<\/mn><\/msub><mo>\u00d7<\/mo><msub><mi>b<\/mi><mn>2<\/mn><\/msub><\/mrow><annotation encoding=\"application\/x-tex\">b_1 \\times b_2<\/annotation><\/semantics><\/math> must be product of <strong>2c<\/strong>:<br><math data-latex=\"2c = 2(-462) = -924\"><semantics><mrow><mn>2<\/mn><mi>c<\/mi><mo>=<\/mo><mn>2<\/mn><mo form=\"prefix\" stretchy=\"false\">(<\/mo><mo form=\"prefix\" stretchy=\"false\">\u2212<\/mo><mn>462<\/mn><mo form=\"postfix\" stretchy=\"false\">)<\/mo><mo>=<\/mo><mo form=\"prefix\" stretchy=\"false\">\u2212<\/mo><mn>924<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">2c = 2(-462) = -924<\/annotation><\/semantics><\/math>, <strong>so<\/strong> <math data-latex=\"(b_1 \\times b_2) = (-28 \\times 33) = -924\"><semantics><mrow><mo form=\"prefix\" stretchy=\"false\">(<\/mo><msub><mi>b<\/mi><mn>1<\/mn><\/msub><mo>\u00d7<\/mo><msub><mi>b<\/mi><mn>2<\/mn><\/msub><mo form=\"postfix\" stretchy=\"false\">)<\/mo><mo>=<\/mo><mo form=\"prefix\" stretchy=\"false\">(<\/mo><mo form=\"prefix\" stretchy=\"false\">\u2212<\/mo><mn>28<\/mn><mo>\u00d7<\/mo><mn>33<\/mn><mo form=\"postfix\" stretchy=\"false\">)<\/mo><mo>=<\/mo><mo form=\"prefix\" stretchy=\"false\">\u2212<\/mo><mn>924<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">(b_1 \\times b_2) = (-28 \\times 33) = -924<\/annotation><\/semantics><\/math><br>Now, let&#8217;s put it all together:<\/p>\n\n\n\n<figure class=\"wp-block-table\"><table class=\"has-fixed-layout\"><thead><tr><th class=\"has-text-align-center\" data-align=\"center\">Formula side<\/th><th class=\"has-text-align-center\" data-align=\"center\">Actual Solving<\/th><\/tr><\/thead><tbody><tr><td class=\"has-text-align-center\" data-align=\"center\"><math data-latex=\"ax^2 + bx + c = 0\"><semantics><mrow><mi>a<\/mi><msup><mi>x<\/mi><mn>2<\/mn><\/msup><mo>+<\/mo><mi>b<\/mi><mi>x<\/mi><mo>+<\/mo><mi>c<\/mi><mo>=<\/mo><mn>0<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">ax^2 + bx + c = 0<\/annotation><\/semantics><\/math><\/td><td class=\"has-text-align-center\" data-align=\"center\"><math data-latex=\"2x^2 + 5x - 462 = 0\"><semantics><mrow><mn>2<\/mn><msup><mi>x<\/mi><mn>2<\/mn><\/msup><mo>+<\/mo><mn>5<\/mn><mi>x<\/mi><mo>\u2212<\/mo><mn>462<\/mn><mo>=<\/mo><mn>0<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">2x^2 + 5x &#8211; 462 = 0<\/annotation><\/semantics><\/math><\/td><\/tr><tr><td class=\"has-text-align-center\" data-align=\"center\"><math data-latex=\"ax^2 + (b_1 + b_2)x + c = 0 \"><semantics><mrow><mi>a<\/mi><msup><mi>x<\/mi><mn>2<\/mn><\/msup><mo>+<\/mo><mo form=\"prefix\" stretchy=\"false\">(<\/mo><msub><mi>b<\/mi><mn>1<\/mn><\/msub><mo>+<\/mo><msub><mi>b<\/mi><mn>2<\/mn><\/msub><mo form=\"postfix\" stretchy=\"false\">)<\/mo><mi>x<\/mi><mo>+<\/mo><mi>c<\/mi><mo>=<\/mo><mn>0<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">ax^2 + (b_1 + b_2)x + c = 0 <\/annotation><\/semantics><\/math><\/td><td class=\"has-text-align-center\" data-align=\"center\"><math data-latex=\"2x^2 + (-28 + 33)x + (-462) = 0 \"><semantics><mrow><mn>2<\/mn><msup><mi>x<\/mi><mn>2<\/mn><\/msup><mo>+<\/mo><mo form=\"prefix\" stretchy=\"false\">(<\/mo><mo form=\"prefix\" stretchy=\"false\">\u2212<\/mo><mn>28<\/mn><mo>+<\/mo><mn>33<\/mn><mo form=\"postfix\" stretchy=\"false\">)<\/mo><mi>x<\/mi><mo>+<\/mo><mo form=\"prefix\" stretchy=\"false\">(<\/mo><mo form=\"prefix\" stretchy=\"false\">\u2212<\/mo><mn>462<\/mn><mo form=\"postfix\" stretchy=\"false\">)<\/mo><mo>=<\/mo><mn>0<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">2x^2 + (-28 + 33)x + (-462) = 0 <\/annotation><\/semantics><\/math><\/td><\/tr><tr><td class=\"has-text-align-center\" data-align=\"center\">Expand<\/td><td class=\"has-text-align-center\" data-align=\"center\"><math data-latex=\"2x^2 -28x + 33x - 462 = 0\"><semantics><mrow><mn>2<\/mn><msup><mi>x<\/mi><mn>2<\/mn><\/msup><mo>\u2212<\/mo><mn>28<\/mn><mi>x<\/mi><mo>+<\/mo><mn>33<\/mn><mi>x<\/mi><mo>\u2212<\/mo><mn>462<\/mn><mo>=<\/mo><mn>0<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">2x^2 -28x + 33x &#8211; 462 = 0<\/annotation><\/semantics><\/math><\/td><\/tr><tr><td class=\"has-text-align-center\" data-align=\"center\"><math data-latex=\"(ax^2 + b_1) + (b_2 + c) = 0\"><semantics><mrow><mo form=\"prefix\" stretchy=\"false\">(<\/mo><mi>a<\/mi><msup><mi>x<\/mi><mn>2<\/mn><\/msup><mo>+<\/mo><msub><mi>b<\/mi><mn>1<\/mn><\/msub><mo form=\"postfix\" stretchy=\"false\">)<\/mo><mo>+<\/mo><mo form=\"prefix\" stretchy=\"false\">(<\/mo><msub><mi>b<\/mi><mn>2<\/mn><\/msub><mo>+<\/mo><mi>c<\/mi><mo form=\"postfix\" stretchy=\"false\">)<\/mo><mo>=<\/mo><mn>0<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">(ax^2 + b_1) + (b_2 + c) = 0<\/annotation><\/semantics><\/math><\/td><td class=\"has-text-align-center\" data-align=\"center\"><math data-latex=\"[2x^2 + (-28x)] + [33x + (-462)] = 0\"><semantics><mrow><mo form=\"prefix\" stretchy=\"false\">[<\/mo><mn>2<\/mn><msup><mi>x<\/mi><mn>2<\/mn><\/msup><mo>+<\/mo><mo form=\"prefix\" stretchy=\"false\">(<\/mo><mo form=\"prefix\" stretchy=\"false\">\u2212<\/mo><mn>28<\/mn><mi>x<\/mi><mo form=\"postfix\" stretchy=\"false\">)<\/mo><mo form=\"postfix\" stretchy=\"false\">]<\/mo><mo>+<\/mo><mo form=\"prefix\" stretchy=\"false\">[<\/mo><mn>33<\/mn><mi>x<\/mi><mo>+<\/mo><mo form=\"prefix\" stretchy=\"false\">(<\/mo><mo form=\"prefix\" stretchy=\"false\">\u2212<\/mo><mn>462<\/mn><mo form=\"postfix\" stretchy=\"false\">)<\/mo><mo form=\"postfix\" stretchy=\"false\">]<\/mo><mo>=<\/mo><mn>0<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">[2x^2 + (-28x)] + [33x + (-462)] = 0<\/annotation><\/semantics><\/math><\/td><\/tr><tr><td class=\"has-text-align-center\" data-align=\"center\">&#8211;<\/td><td class=\"has-text-align-center\" data-align=\"center\"><math data-latex=\"(2x^2 - 28x) + (33x - 462) = 0\"><semantics><mrow><mo form=\"prefix\" stretchy=\"false\">(<\/mo><mn>2<\/mn><msup><mi>x<\/mi><mn>2<\/mn><\/msup><mo>\u2212<\/mo><mn>28<\/mn><mi>x<\/mi><mo form=\"postfix\" stretchy=\"false\">)<\/mo><mo>+<\/mo><mo form=\"prefix\" stretchy=\"false\">(<\/mo><mn>33<\/mn><mi>x<\/mi><mo>\u2212<\/mo><mn>462<\/mn><mo form=\"postfix\" stretchy=\"false\">)<\/mo><mo>=<\/mo><mn>0<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">(2x^2 &#8211; 28x) + (33x &#8211; 462) = 0<\/annotation><\/semantics><\/math><\/td><\/tr><tr><td class=\"has-text-align-center\" data-align=\"center\">Rewrite to make Common<\/td><td class=\"has-text-align-center\" data-align=\"center\"><math data-latex=\"2x(x - 14x) + 33(x - 14) = 0\"><semantics><mrow><mn>2<\/mn><mi>x<\/mi><mo form=\"prefix\" stretchy=\"false\">(<\/mo><mi>x<\/mi><mo>\u2212<\/mo><mn>14<\/mn><mi>x<\/mi><mo form=\"postfix\" stretchy=\"false\">)<\/mo><mo>+<\/mo><mn>33<\/mn><mo form=\"prefix\" stretchy=\"false\">(<\/mo><mi>x<\/mi><mo>\u2212<\/mo><mn>14<\/mn><mo form=\"postfix\" stretchy=\"false\">)<\/mo><mo>=<\/mo><mn>0<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">2x(x &#8211; 14x) + 33(x &#8211; 14) = 0<\/annotation><\/semantics><\/math><\/td><\/tr><\/tbody><tfoot><tr><td class=\"has-text-align-center\" data-align=\"center\">Common value out<\/td><td class=\"has-text-align-center\" data-align=\"center\"><math data-latex=\"(2x + 33)(x - 14x) = 0\"><semantics><mrow><mo form=\"prefix\" stretchy=\"false\">(<\/mo><mn>2<\/mn><mi>x<\/mi><mo>+<\/mo><mn>33<\/mn><mo form=\"postfix\" stretchy=\"false\">)<\/mo><mo form=\"prefix\" stretchy=\"false\">(<\/mo><mi>x<\/mi><mo>\u2212<\/mo><mn>14<\/mn><mi>x<\/mi><mo form=\"postfix\" stretchy=\"false\">)<\/mo><mo>=<\/mo><mn>0<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">(2x + 33)(x &#8211; 14x) = 0<\/annotation><\/semantics><\/math><\/td><\/tr><\/tfoot><\/table><\/figure>\n\n\n\n<p class=\"is-style-success\" style=\"font-size:0.9em\">Factor the equation:<math display=\"block\"><semantics><mrow><mo stretchy=\"false\">(<\/mo><mn>2<\/mn><mi>x<\/mi><mo>+<\/mo><mn>33<\/mn><mo stretchy=\"false\">)<\/mo><mo stretchy=\"false\">(<\/mo><mi>x<\/mi><mo>\u2212<\/mo><mn>14<\/mn><mo stretchy=\"false\">)<\/mo><mo>=<\/mo><mn>0<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">(2x + 33)(x &#8211; 14) = 0<\/annotation><\/semantics><\/math><br><math data-latex=\"(2x + 33) = 0\"><semantics><mrow><mo stretchy=\"false\" form=\"prefix\">(<\/mo><mn>2<\/mn><mi>x<\/mi><mo>+<\/mo><mn>33<\/mn><mo stretchy=\"false\" form=\"postfix\">)<\/mo><mo>=<\/mo><mn>0<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">(2x + 33) = 0<\/annotation><\/semantics><\/math> then <math data-latex=\"2x + 33 = 0\"><semantics><mrow><mn>2<\/mn><mi>x<\/mi><mo>+<\/mo><mn>33<\/mn><mo>=<\/mo><mn>0<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">2x + 33 = 0<\/annotation><\/semantics><\/math> then <math data-latex=\"2x = -33\"><semantics><mrow><mn>2<\/mn><mi>x<\/mi><mo>=<\/mo><mo form=\"prefix\" stretchy=\"false\">\u2212<\/mo><mn>33<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">2x = -33<\/annotation><\/semantics><\/math> then <math data-latex=\"x = - \\frac{33}{2}\"><semantics><mrow><mi>x<\/mi><mo>=<\/mo><mo form=\"prefix\" stretchy=\"false\">\u2212<\/mo><mfrac><mn>33<\/mn><mn>2<\/mn><\/mfrac><\/mrow><annotation encoding=\"application\/x-tex\">x = &#8211; \\frac{33}{2}<\/annotation><\/semantics><\/math><br><math data-latex=\"(x - 14) = 0\"><semantics><mrow><mo form=\"prefix\" stretchy=\"false\">(<\/mo><mi>x<\/mi><mo>\u2212<\/mo><mn>14<\/mn><mo form=\"postfix\" stretchy=\"false\">)<\/mo><mo>=<\/mo><mn>0<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">(x &#8211; 14) = 0<\/annotation><\/semantics><\/math> then <math data-latex=\"x - 14 = 0\"><semantics><mrow><mi>x<\/mi><mo>\u2212<\/mo><mn>14<\/mn><mo>=<\/mo><mn>0<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">x &#8211; 14 = 0<\/annotation><\/semantics><\/math> then <math data-latex=\"x = 14\"><semantics><mrow><mi>x<\/mi><mo>=<\/mo><mn>14<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">x = 14<\/annotation><\/semantics><\/math><br><br>So:<math display=\"block\"><semantics><mrow><mi>x<\/mi><mo>=<\/mo><mo>\u2212<\/mo><mfrac><mn>33<\/mn><mn>2<\/mn><\/mfrac><mspace width=\"1em\"><\/mspace><mtext>or<\/mtext><mspace width=\"1em\"><\/mspace><mi>x<\/mi><mo>=<\/mo><mn>14<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">x = -\\frac{33}{2} \\quad \\text{or} \\quad x = 14<\/annotation><\/semantics><\/math><br><strong>Method 2:<br><\/strong><math data-latex=\"\\\\ 2x^2 + 5x - 462 = 0\"><semantics><mtable columnalign=\"left\" rowspacing=\"0em\"><mtr><mtd style=\"text-align:left\"><mo><\/mo><\/mtd><\/mtr><mtr><mtd style=\"text-align:left\"><mrow><mn>2<\/mn><msup><mi>x<\/mi><mn>2<\/mn><\/msup><mo>+<\/mo><mn>5<\/mn><mi>x<\/mi><mo>\u2212<\/mo><mn>462<\/mn><mo>=<\/mo><mn>0<\/mn><\/mrow><\/mtd><\/mtr><\/mtable><annotation encoding=\"application\/x-tex\">\\\\ 2x^2 + 5x &#8211; 462 = 0<\/annotation><\/semantics><\/math><br><br>The formula: <math data-latex=\"x=\\frac{-b\\pm \\sqrt{b^2 - 4ac}}{2a}\"><semantics><mrow><mi>x<\/mi><mo>=<\/mo><mfrac><mrow><mo lspace=\"0em\" rspace=\"0em\">\u2212<\/mo><mi>b<\/mi><mo>\u00b1<\/mo><msqrt><mrow><msup><mi>b<\/mi><mn>2<\/mn><\/msup><mo>\u2212<\/mo><mn>4<\/mn><mi>a<\/mi><mi>c<\/mi><\/mrow><\/msqrt><\/mrow><mrow><mn>2<\/mn><mi>a<\/mi><\/mrow><\/mfrac><\/mrow><annotation encoding=\"application\/x-tex\">x=\\frac{-b\\pm \\sqrt{b^2 &#8211; 4ac}}{2a}<\/annotation><\/semantics><\/math><br><br>Put values: <math data-latex=\"x=\\frac{-5\\pm \\sqrt{5^2-4 \\times 2 \\times -462}}{2 \\times 2}\"><semantics><mrow><mi>x<\/mi><mo>=<\/mo><mfrac><mrow><mo lspace=\"0em\" rspace=\"0em\">\u2212<\/mo><mn>5<\/mn><mo>\u00b1<\/mo><msqrt><mrow><msup><mn>5<\/mn><mn>2<\/mn><\/msup><mo>\u2212<\/mo><mn>4<\/mn><mo>\u00d7<\/mo><mn>2<\/mn><mo>\u00d7<\/mo><mo form=\"prefix\" stretchy=\"false\">\u2212<\/mo><mn>462<\/mn><\/mrow><\/msqrt><\/mrow><mrow><mn>2<\/mn><mo>\u00d7<\/mo><mn>2<\/mn><\/mrow><\/mfrac><\/mrow><annotation encoding=\"application\/x-tex\">x=\\frac{-5\\pm \\sqrt{5^2-4 \\times 2 \\times -462}}{2 \\times 2}<\/annotation><\/semantics><\/math><br><br><math data-latex=\"x=\\frac{-5\\pm \\sqrt{25 + 3696}}{4}\"><semantics><mrow><mi>x<\/mi><mo>=<\/mo><mfrac><mrow><mo lspace=\"0em\" rspace=\"0em\">\u2212<\/mo><mn>5<\/mn><mo>\u00b1<\/mo><msqrt><mrow><mn>25<\/mn><mo>+<\/mo><mn>3696<\/mn><\/mrow><\/msqrt><\/mrow><mn>4<\/mn><\/mfrac><\/mrow><annotation encoding=\"application\/x-tex\">x=\\frac{-5\\pm \\sqrt{25 + 3696}}{4}<\/annotation><\/semantics><\/math><br><br><math data-latex=\"x=\\frac{-5\\pm \\sqrt{3721}}{4}\"><semantics><mrow><mi>x<\/mi><mo>=<\/mo><mfrac><mrow><mo lspace=\"0em\" rspace=\"0em\">\u2212<\/mo><mn>5<\/mn><mo>\u00b1<\/mo><msqrt><mn>3721<\/mn><\/msqrt><\/mrow><mn>4<\/mn><\/mfrac><\/mrow><annotation encoding=\"application\/x-tex\">x=\\frac{-5\\pm \\sqrt{3721}}{4}<\/annotation><\/semantics><\/math><br><br><math data-latex=\"x=\\frac{-5\\pm 61}{4}\"><semantics><mrow><mi>x<\/mi><mo>=<\/mo><mfrac><mrow><mo lspace=\"0em\" rspace=\"0em\">\u2212<\/mo><mn>5<\/mn><mo>\u00b1<\/mo><mn>61<\/mn><\/mrow><mn>4<\/mn><\/mfrac><\/mrow><annotation encoding=\"application\/x-tex\">x=\\frac{-5\\pm 61}{4}<\/annotation><\/semantics><\/math><br><br><math data-latex=\"\\pm\"><semantics><mo lspace=\"0em\" rspace=\"0em\">\u00b1<\/mo><annotation encoding=\"application\/x-tex\">\\pm<\/annotation><\/semantics><\/math> this sign means, we can do either plus or minus, so:<br><br><math data-latex=\"x=\\frac{-5 +  61}{4}\"><semantics><mrow><mi>x<\/mi><mo>=<\/mo><mfrac><mrow><mo lspace=\"0em\" rspace=\"0em\">\u2212<\/mo><mn>5<\/mn><mo>+<\/mo><mn>61<\/mn><\/mrow><mn>4<\/mn><\/mfrac><\/mrow><annotation encoding=\"application\/x-tex\">x=\\frac{-5 +  61}{4}<\/annotation><\/semantics><\/math>  or  <math data-latex=\"x=\\frac{-5 -  61}{4}\"><semantics><mrow><mi>x<\/mi><mo>=<\/mo><mfrac><mrow><mo lspace=\"0em\" rspace=\"0em\">\u2212<\/mo><mn>5<\/mn><mo>\u2212<\/mo><mn>61<\/mn><\/mrow><mn>4<\/mn><\/mfrac><\/mrow><annotation encoding=\"application\/x-tex\">x=\\frac{-5 &#8211;  61}{4}<\/annotation><\/semantics><\/math><br><br><math data-latex=\"x=\\frac{56}{4}\"><semantics><mrow><mi>x<\/mi><mo>=<\/mo><mfrac><mn>56<\/mn><mn>4<\/mn><\/mfrac><\/mrow><annotation encoding=\"application\/x-tex\">x=\\frac{56}{4}<\/annotation><\/semantics><\/math>  or  <math data-latex=\"x=\\frac{-66}{4}\"><semantics><mrow><mi>x<\/mi><mo>=<\/mo><mfrac><mrow><mo lspace=\"0em\" rspace=\"0em\">\u2212<\/mo><mn>66<\/mn><\/mrow><mn>4<\/mn><\/mfrac><\/mrow><annotation encoding=\"application\/x-tex\">x=\\frac{-66}{4}<\/annotation><\/semantics><\/math><br><br><math data-latex=\"x=14\"><semantics><mrow><mi>x<\/mi><mo>=<\/mo><mn>14<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">x=14<\/annotation><\/semantics><\/math>  or  <math data-latex=\"x= - \\frac{33}{2}\"><semantics><mrow><mi>x<\/mi><mo>=<\/mo><mo form=\"prefix\" stretchy=\"false\">\u2212<\/mo><mfrac><mn>33<\/mn><mn>2<\/mn><\/mfrac><\/mrow><annotation encoding=\"application\/x-tex\">x= &#8211; \\frac{33}{2}<\/annotation><\/semantics><\/math><br><br><strong>Both Method 1 and 2 are correct, choose according to your choice.<br><\/strong><br><strong>\ud83d\udeab STEP 3: APPLY CONTEXT RESTRICTIONS<\/strong><br>The question states <strong>positive integers<\/strong>, so:<br>~ Fractions are <strong>not allowed<\/strong><br>~ Negative values are <strong>not allowed<\/strong><br>~ and we need a <strong>smaller integer<\/strong><br>Thus:<math display=\"block\"><semantics><mrow><mi>x<\/mi><mo>=<\/mo><mn>14<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">x = 14<\/annotation><\/semantics><\/math><br><strong><span class=\"uppercase\">Verifying:<br><\/span><\/strong>1. Input <strong>x = 14 <\/strong>into the equation: <math data-latex=\"x(2x + 5) = 462\"><semantics><mrow><mi>x<\/mi><mo form=\"prefix\" stretchy=\"false\">(<\/mo><mn>2<\/mn><mi>x<\/mi><mo>+<\/mo><mn>5<\/mn><mo form=\"postfix\" stretchy=\"false\">)<\/mo><mo>=<\/mo><mn>462<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">x(2x + 5) = 462<\/annotation><\/semantics><\/math><br><math data-latex=\"\\\\ 14(2 \\times 14 + 5) = 462\\\\ \\\\14(28 + 5) = 462\\\\ \\\\14(33) = 462\\\\ \\\\462 = 462\"><semantics><mtable columnalign=\"left\" rowspacing=\"0em\"><mtr><mtd style=\"text-align:left\"><mo><\/mo><\/mtd><\/mtr><mtr><mtd style=\"text-align:left\"><mrow><mn>14<\/mn><mo form=\"prefix\" stretchy=\"false\">(<\/mo><mn>2<\/mn><mo>\u00d7<\/mo><mn>14<\/mn><mo>+<\/mo><mn>5<\/mn><mo form=\"postfix\" stretchy=\"false\">)<\/mo><mo>=<\/mo><mn>462<\/mn><\/mrow><mo><\/mo><\/mtd><\/mtr><mtr><mtd style=\"text-align:left\"><mo><\/mo><\/mtd><\/mtr><mtr><mtd style=\"text-align:left\"><mrow><mn>14<\/mn><mo form=\"prefix\" stretchy=\"false\">(<\/mo><mn>28<\/mn><mo>+<\/mo><mn>5<\/mn><mo form=\"postfix\" stretchy=\"false\">)<\/mo><mo>=<\/mo><mn>462<\/mn><\/mrow><mo><\/mo><\/mtd><\/mtr><mtr><mtd style=\"text-align:left\"><mo><\/mo><\/mtd><\/mtr><mtr><mtd style=\"text-align:left\"><mrow><mn>14<\/mn><mo form=\"prefix\" stretchy=\"false\">(<\/mo><mn>33<\/mn><mo form=\"postfix\" stretchy=\"false\">)<\/mo><mo>=<\/mo><mn>462<\/mn><\/mrow><mo><\/mo><\/mtd><\/mtr><mtr><mtd style=\"text-align:left\"><mo><\/mo><\/mtd><\/mtr><mtr><mtd style=\"text-align:left\"><mrow><mn>462<\/mn><mo>=<\/mo><mn>462<\/mn><\/mrow><\/mtd><\/mtr><\/mtable><annotation encoding=\"application\/x-tex\">\\\\ 14(2 \\times 14 + 5) = 462\\\\ \\\\14(28 + 5) = 462\\\\ \\\\14(33) = 462\\\\ \\\\462 = 462<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<p class=\"is-style-error\" style=\"font-size:0.9em\"><strong>\u26a0\ufe0f COMMON TRAPS (REAL MISTAKES)<br><\/strong>\u274c Forgetting integers must be whole numbers<br>\u274c Forgetting \u201ctwice\u201d and writing <math><semantics><mrow><mi>x<\/mi><mo>+<\/mo><mn>5<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">x + 5<\/annotation><\/semantics><\/math><br>\u274c Solving the quadratic but choosing the <strong>negative root<\/strong><br>\u274c Giving the <strong>larger integer<\/strong> instead of the smaller<\/p>\n\n\n\n<p class=\"is-style-info\" style=\"font-size:0.9em\"><strong>\ud83e\uddee DESMOS METHOD (REAL &amp; VALID)<br><\/strong>Step 1: Open Desmos<br>Type: x(2x + 5) = 462<br>Step 2: Convert to y-form<br>y = x(2x + 5) &#8211; 462<br>Step 3: Find x-intercepts<br>Desmos shows: Go to graph and only focus on x-axis<br>x \u2248 \u221216.5<br>x = 14<br>Only <strong>14<\/strong> satisfies the conditions.<\/p>\n<\/div><\/details><\/div>\n\n\n\n<div class=\"wp-block-coblocks-accordion-item\"><details><summary class=\"wp-block-coblocks-accordion-item__title\"><strong>22th Question<\/strong><\/summary><div class=\"wp-block-coblocks-accordion-item__content\">\n<p class=\"is-style-warning\" style=\"font-size:0.9em\"><strong>Question:<\/strong> The function <math data-latex=\"k\"><semantics><mi>k<\/mi><annotation encoding=\"application\/x-tex\">k<\/annotation><\/semantics><\/math> is defined by <math data-latex=\"k(x) = \\frac{|x|}{a} - 14\"><semantics><mrow><mi>k<\/mi><mo form=\"prefix\" stretchy=\"false\">(<\/mo><mi>x<\/mi><mo form=\"postfix\" stretchy=\"false\">)<\/mo><mo>=<\/mo><mfrac><mrow><mi>|<\/mi><mi>x<\/mi><mi>|<\/mi><\/mrow><mi>a<\/mi><\/mfrac><mo>\u2212<\/mo><mn>14<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">k(x) = \\frac{|x|}{a} &#8211; 14<\/annotation><\/semantics><\/math>, where <math data-latex=\"a &lt; 0\"><semantics><mrow><mi>a<\/mi><mo>&lt;<\/mo><mn>0<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">a &lt; 0<\/annotation><\/semantics><\/math>. What is the product of <math data-latex=\"k(15a)\"><semantics><mrow><mi>k<\/mi><mo form=\"prefix\" stretchy=\"false\">(<\/mo><mn>15<\/mn><mi>a<\/mi><mo form=\"postfix\" stretchy=\"false\">)<\/mo><\/mrow><annotation encoding=\"application\/x-tex\">k(15a)<\/annotation><\/semantics><\/math> and <math data-latex=\"k(7a)\"><semantics><mrow><mi>k<\/mi><mo form=\"prefix\" stretchy=\"false\">(<\/mo><mn>7<\/mn><mi>a<\/mi><mo form=\"postfix\" stretchy=\"false\">)<\/mo><\/mrow><annotation encoding=\"application\/x-tex\">k(7a)<\/annotation><\/semantics><\/math>?<br><br>[Type-Based Answer: In the final exam, you will type the answer rather than choose from options.]<\/p>\n\n\n\n<p class=\"is-style-success\"><strong>Given<\/strong><br><math display=\"block\"><semantics><mrow><mi>k<\/mi><mo stretchy=\"false\">(<\/mo><mi>x<\/mi><mo stretchy=\"false\">)<\/mo><mo>=<\/mo><mfrac><mrow><mi mathvariant=\"normal\">\u2223<\/mi><mi>x<\/mi><mi mathvariant=\"normal\">\u2223<\/mi><\/mrow><mi>a<\/mi><\/mfrac><mo>\u2212<\/mo><mn>14<\/mn><mspace width=\"1em\"><\/mspace><mtext>where&nbsp;<\/mtext><mi>a<\/mi><mo>&lt;<\/mo><mn>0<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">k(x) = \\frac{|x|}{a} &#8211; 14 \\quad \\text{where } a &lt; 0<\/annotation><\/semantics><\/math><br>Find:<math display=\"block\"><semantics><mrow><mi>k<\/mi><mo stretchy=\"false\">(<\/mo><mn>15<\/mn><mi>a<\/mi><mo stretchy=\"false\">)<\/mo><mo>\u00d7<\/mo><mi>k<\/mi><mo stretchy=\"false\">(<\/mo><mn>7<\/mn><mi>a<\/mi><mo stretchy=\"false\">)<\/mo><\/mrow><annotation encoding=\"application\/x-tex\">k(15a) \\times k(7a)<\/annotation><\/semantics><\/math><br><strong>\u2705 Step-by-Step Mathematical Solution<\/strong><br><strong>\ud83d\udd39 Step 1: Understand the key conditions<\/strong><br><br>~ If <math data-latex=\"k(x) = \\frac{|x|}{a} - 14\"><semantics><mrow><mi>k<\/mi><mo form=\"prefix\" stretchy=\"false\">(<\/mo><mi>x<\/mi><mo form=\"postfix\" stretchy=\"false\">)<\/mo><mo>=<\/mo><mfrac><mrow><mi>|<\/mi><mi>x<\/mi><mi>|<\/mi><\/mrow><mi>a<\/mi><\/mfrac><mo>\u2212<\/mo><mn>14<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">k(x) = \\frac{|x|}{a} &#8211; 14<\/annotation><\/semantics><\/math>, then:<br> <math data-latex=\"k(x) = k(15a)\"><semantics><mrow><mi>k<\/mi><mo form=\"prefix\" stretchy=\"false\">(<\/mo><mi>x<\/mi><mo form=\"postfix\" stretchy=\"false\">)<\/mo><mo>=<\/mo><mi>k<\/mi><mo form=\"prefix\" stretchy=\"false\">(<\/mo><mn>15<\/mn><mi>a<\/mi><mo form=\"postfix\" stretchy=\"false\">)<\/mo><\/mrow><annotation encoding=\"application\/x-tex\">k(x) = k(15a)<\/annotation><\/semantics><\/math>  and  <math data-latex=\"k(7a)\"><semantics><mrow><mi>k<\/mi><mo form=\"prefix\" stretchy=\"false\">(<\/mo><mn>7<\/mn><mi>a<\/mi><mo form=\"postfix\" stretchy=\"false\">)<\/mo><\/mrow><annotation encoding=\"application\/x-tex\">k(7a)<\/annotation><\/semantics><\/math>, it means<br><math data-latex=\"x = 15a\"><semantics><mrow><mi>x<\/mi><mo>=<\/mo><mn>15<\/mn><mi>a<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">x = 15a<\/annotation><\/semantics><\/math>  and  <math data-latex=\"7a\"><semantics><mrow><mn>7<\/mn><mi>a<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">7a<\/annotation><\/semantics><\/math><br><br>Recall the <strong>absolute value rule<\/strong>:<br>This condition controls how <strong>absolute value behaves<\/strong>.<br><math display=\"block\"><semantics><mrow><mi mathvariant=\"normal\">\u2223<\/mi><mi>u<\/mi><mi mathvariant=\"normal\">\u2223<\/mi><mo>=<\/mo><mrow><mo fence=\"true\">{<\/mo><mtable rowspacing=\"0.36em\" columnalign=\"left left\" columnspacing=\"1em\"><mtr><mtd><mstyle scriptlevel=\"0\" displaystyle=\"false\"><mrow><mi>u<\/mi><mo separator=\"true\">,<\/mo><\/mrow><\/mstyle><\/mtd><mtd><mstyle scriptlevel=\"0\" displaystyle=\"false\"><mrow><mi>u<\/mi><mo>\u2265<\/mo><mn>0<\/mn><\/mrow><\/mstyle><\/mtd><\/mtr><mtr><mtd><mstyle scriptlevel=\"0\" displaystyle=\"false\"><mrow><mo>\u2212<\/mo><mi>u<\/mi><mo separator=\"true\">,<\/mo><\/mrow><\/mstyle><\/mtd><mtd><mstyle scriptlevel=\"0\" displaystyle=\"false\"><mrow><mi>u<\/mi><mo>&lt;<\/mo><mn>0<\/mn><\/mrow><\/mstyle><\/mtd><\/mtr><\/mtable><\/mrow><\/mrow><annotation encoding=\"application\/x-tex\">|u| = \\begin{cases} u, &amp; u \\ge 0 \\\\ &#8211; u, &amp; u &lt; 0 \\end{cases}<\/annotation><\/semantics><\/math><br><strong>u<\/strong> is just to demonstrate, you can replace it with <strong>a<\/strong> or anything else.<br>~ If <math><semantics><mrow><mi>a<\/mi><mo>&lt;<\/mo><mn>0<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">a &lt; 0<\/annotation><\/semantics><\/math>, then: <math data-latex=\"|a| = -a\"><semantics><mrow><mi>|<\/mi><mi>a<\/mi><mi>|<\/mi><mo>=<\/mo><mo form=\"prefix\" stretchy=\"false\">\u2212<\/mo><mi>a<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">|a| = -a<\/annotation><\/semantics><\/math><br><math data-latex=\"15a\"><semantics><mrow><mn>15<\/mn><mi>a<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">15a<\/annotation><\/semantics><\/math>  and  <math data-latex=\"7a\"><semantics><mrow><mn>7<\/mn><mi>a<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">7a<\/annotation><\/semantics><\/math><br>So the <math data-latex=\"a\"><semantics><mi>a<\/mi><annotation encoding=\"application\/x-tex\">a<\/annotation><\/semantics><\/math> in both inputs is a <strong>negative numbers<\/strong>.<br><br><strong>\ud83d\udd39 Step 2: Evaluate <\/strong><math><semantics><mrow><mi>k<\/mi><mo stretchy=\"false\">(<\/mo><mn>15<\/mn><mi>a<\/mi><mo stretchy=\"false\">)<\/mo><\/mrow><annotation encoding=\"application\/x-tex\">k(15a)<\/annotation><\/semantics><\/math><br>Start with the definition:<math display=\"block\"><semantics><mrow><mi>k<\/mi><mo stretchy=\"false\">(<\/mo><mn>15<\/mn><mi>a<\/mi><mo stretchy=\"false\">)<\/mo><mo>=<\/mo><mfrac><mrow><mi mathvariant=\"normal\">\u2223<\/mi><mn>15<\/mn><mi>a<\/mi><mi mathvariant=\"normal\">\u2223<\/mi><\/mrow><mi>a<\/mi><\/mfrac><mo>\u2212<\/mo><mn>14<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">k(15a) = \\frac{|15a|}{a} &#8211; 14<\/annotation><\/semantics><\/math><br>Absolute Value Handling (Critical Concept)<br>Since:<br>15 is positive<br><math><semantics><mrow><mi>a<\/mi><mo>&lt;<\/mo><mn>0<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">a &lt; 0<\/annotation><\/semantics><\/math><br>Then:<math display=\"block\"><semantics><mrow><mi mathvariant=\"normal\">\u2223<\/mi><mn>15<\/mn><mi>a<\/mi><mi mathvariant=\"normal\">\u2223<\/mi><mo>=<\/mo><mn>15<\/mn><mi mathvariant=\"normal\">\u2223<\/mi><mi>a<\/mi><mi mathvariant=\"normal\">\u2223<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">|15a| = 15|a|<\/annotation><\/semantics><\/math><br>And because <math><semantics><mrow><mi>a<\/mi><mo>&lt;<\/mo><mn>0<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">a &lt; 0<\/annotation><\/semantics><\/math>, we know:<math display=\"block\"><semantics><mrow><mi mathvariant=\"normal\">\u2223<\/mi><mi>a<\/mi><mi mathvariant=\"normal\">\u2223<\/mi><mo>=<\/mo><mo>\u2212<\/mo><mi>a<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">|a| = -a<\/annotation><\/semantics><\/math><br>So:<math display=\"block\"><semantics><mrow><mi mathvariant=\"normal\">\u2223<\/mi><mn>15<\/mn><mi>a<\/mi><mi mathvariant=\"normal\">\u2223<\/mi><mo>=<\/mo><mn>15<\/mn><mo stretchy=\"false\">(<\/mo><mo>\u2212<\/mo><mi>a<\/mi><mo stretchy=\"false\">)<\/mo><mo>=<\/mo><mo>\u2212<\/mo><mn>15<\/mn><mi>a<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">|15a| = 15(-a) = -15a<\/annotation><\/semantics><\/math><br>Substitute back:<math display=\"block\"><semantics><mrow><mi>k<\/mi><mo stretchy=\"false\">(<\/mo><mn>15<\/mn><mi>a<\/mi><mo stretchy=\"false\">)<\/mo><mo>=<\/mo><mfrac><mrow><mo>\u2212<\/mo><mn>15<\/mn><mi>a<\/mi><\/mrow><mi>a<\/mi><\/mfrac><mo>\u2212<\/mo><mn>14<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">k(15a) = \\frac{-15a}{a} &#8211; 14<\/annotation><\/semantics><\/math><br>Cancel <math><semantics><mrow><mi>a<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">a<\/annotation><\/semantics><\/math> (allowed since <math><semantics><mrow><mi>a<\/mi><mo>\u2260<\/mo><mn>0<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">a \\neq 0<\/annotation><\/semantics><\/math>):<br><math display=\"block\"><semantics><mrow><mi>k<\/mi><mo stretchy=\"false\">(<\/mo><mn>15<\/mn><mi>a<\/mi><mo stretchy=\"false\">)<\/mo><mo>=<\/mo><mo>\u2212<\/mo><mn>15<\/mn><mo>\u2212<\/mo><mn>14<\/mn><mo>=<\/mo><mo>\u2212<\/mo><mn>29<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">k(15a) = -15 &#8211; 14 = -29<\/annotation><\/semantics><\/math><br><br><strong>\ud83d\udd39 Step 3: Evaluate <\/strong><math><semantics><mrow><mi>k<\/mi><mo stretchy=\"false\">(<\/mo><mn>7<\/mn><mi>a<\/mi><mo stretchy=\"false\">)<\/mo><\/mrow><annotation encoding=\"application\/x-tex\">k(7a)<\/annotation><\/semantics><\/math><br>Same logic, new input:<math display=\"block\"><semantics><mrow><mi>k<\/mi><mo stretchy=\"false\">(<\/mo><mn>7<\/mn><mi>a<\/mi><mo stretchy=\"false\">)<\/mo><mo>=<\/mo><mfrac><mrow><mi mathvariant=\"normal\">\u2223<\/mi><mn>7<\/mn><mi>a<\/mi><mi mathvariant=\"normal\">\u2223<\/mi><\/mrow><mi>a<\/mi><\/mfrac><mo>\u2212<\/mo><mn>14<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">k(7a) = \\frac{|7a|}{a} &#8211; 14<\/annotation><\/semantics><\/math><br>Because <math><semantics><mrow><mi>a<\/mi><mo>&lt;<\/mo><mn>0<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">a &lt; 0<\/annotation><\/semantics><\/math>:<math display=\"block\"><semantics><mrow><mi mathvariant=\"normal\">\u2223<\/mi><mn>7<\/mn><mi>a<\/mi><mi mathvariant=\"normal\">\u2223<\/mi><mo>=<\/mo><mo>\u2212<\/mo><mn>7<\/mn><mi>a<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">|7a| = -7a<\/annotation><\/semantics><\/math><br>So:<math display=\"block\"><semantics><mrow><mi>k<\/mi><mo stretchy=\"false\">(<\/mo><mn>7<\/mn><mi>a<\/mi><mo stretchy=\"false\">)<\/mo><mo>=<\/mo><mfrac><mrow><mo>\u2212<\/mo><mn>7<\/mn><mi>a<\/mi><\/mrow><mi>a<\/mi><\/mfrac><mo>\u2212<\/mo><mn>14<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">k(7a) = \\frac{-7a}{a} &#8211; 14<\/annotation><\/semantics><\/math><br>Cancel <math><semantics><mrow><mi>a<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">a<\/annotation><\/semantics><\/math>:<math display=\"block\"><semantics><mrow><mi>k<\/mi><mo stretchy=\"false\">(<\/mo><mn>7<\/mn><mi>a<\/mi><mo stretchy=\"false\">)<\/mo><mo>=<\/mo><mo>\u2212<\/mo><mn>7<\/mn><mo>\u2212<\/mo><mn>14<\/mn><mo>=<\/mo><mo>\u2212<\/mo><mn>21<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">k(7a) = -7 &#8211; 14 = -21<\/annotation><\/semantics><\/math><br><strong>\ud83d\udd39 Step 4: Find the Product<br><\/strong><math display=\"block\"><semantics><mrow><mi>k<\/mi><mo stretchy=\"false\">(<\/mo><mn>15<\/mn><mi>a<\/mi><mo stretchy=\"false\">)<\/mo><mo>\u00d7<\/mo><mi>k<\/mi><mo stretchy=\"false\">(<\/mo><mn>7<\/mn><mi>a<\/mi><mo stretchy=\"false\">)<\/mo><mo>=<\/mo><mo stretchy=\"false\">(<\/mo><mo>\u2212<\/mo><mn>29<\/mn><mo stretchy=\"false\">)<\/mo><mo stretchy=\"false\">(<\/mo><mo>\u2212<\/mo><mn>21<\/mn><mo stretchy=\"false\">)<\/mo><\/mrow><annotation encoding=\"application\/x-tex\">k(15a) \\times k(7a) = (-29)(-21)<\/annotation><\/semantics><\/math><br><math display=\"block\"><semantics><mrow><mo>=<\/mo><mn>609<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">= 609<\/annotation><\/semantics><\/math><br><strong>\u2705 Final Answer: 609<\/strong><\/p>\n\n\n\n<p class=\"is-style-error\" style=\"font-size:0.9em\">\u274c Trap 1: Treating <code>15a<\/code> and <code>7a<\/code> as \u201cjust a\u201d<br>Wrong \u2014 they are <strong>inputs<\/strong>, not constants.<\/p>\n\n\n\n<p class=\"is-style-error\" style=\"font-size:0.9em\">\u274c Trap 2: Forgetting <code>a &lt; 0<\/code><br>If you assume <code>a &gt; 0<\/code>, you will get <strong>completely wrong signs<\/strong>.<\/p>\n\n\n\n<p class=\"is-style-error\" style=\"font-size:0.9em\">\u274c Trap 3: Dropping absolute value incorrectly<br>Absolute value changes <strong>everything<\/strong> when negatives are involved.<\/p>\n\n\n\n<p class=\"is-style-info\" style=\"font-size:0.9em\">\ud83e\uddee <strong>REAL Desmos Method (Correct Usage Only)<\/strong><br>This is how a student would <strong>actually verify<\/strong> the answer in Desmos.<br><strong>Step 1: Define the function (using a slider)<br><\/strong>1. Type: k(x) = abs(x)\/a &#8211; 14<br>2. Create a <strong>slider for <code>a<\/code><\/strong> and set:<br><code>a = -1<\/code> (or any negative value)<br>~ It means, type any value the answer will come same here but it must be negative.<br>SAT allows this because the expression depends on sign, not magnitude.<br><br><strong>Step 2: Evaluate the function values<br><\/strong>1. In Desmos, type:<br>k(15a)<br>k(7a)<br>2. Desmos will display:<br>k(15a) = -29<br>k(7a) = -21<br><br><strong>Step 3: Multiply directly<br><\/strong>1. Type: k(15a) * k(7a)<br>or<br>just directly type: -29 * -21<br>2. Desmos output: 609<br>\u2705 Confirms the algebraic solution perfectly.<\/p>\n<\/div><\/details><\/div>\n<\/div>\n\n\n\n<div style=\"height:70px\" aria-hidden=\"true\" class=\"wp-block-spacer\"><\/div>\n\n\n\n<p>Did you try all the features and get comfortable using them? You should work on using the calculator and seeing references and directions. So be prepared for everything before taking the final SAT exam. The explanation of answers makes easier to learn and progress. You must try to work on your speed and spend less time on the beginning and more on the later questions. This is the SAT 2024 Practice Test of Math Module 2nd.<\/p>\n\n\n\n<p>There are more tests available:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li><a href=\"https:\/\/us.mrenglishkj.com\/sat\/sat-math-test-4-module-2nd-preparation\/\" target=\"_blank\" rel=\"noopener\" title=\"\">SAT 2025 Test (Math Module 1st)<\/a><\/li>\n\n\n\n<li><a href=\"https:\/\/us.mrenglishkj.com\/sat\/sat-math-test-3-module-1st-study-guide\/\" target=\"_blank\" rel=\"noopener\" title=\"\">SAT Test 3rd (Math Module 1st)<\/a><\/li>\n\n\n\n<li><a href=\"https:\/\/us.mrenglishkj.com\/sat\/sat-reading-and-writing-test-6-module-2nd\/\" target=\"_blank\" rel=\"noopener\" title=\"\">SAT Test 6th (Reading and Writing Module 2nd)<\/a><\/li>\n<\/ul>\n\n\n\n<p>The best way to become a master in Math is to find the correct answer and understand why other options are incorrect. I wish you luck in your bright career.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>2026 SAT Math Test (Free Practice of Module 2 New Questions: Latest questions and their step-by-step explanations of correct and incorrect answers of the SAT math 2026 test with Desmos hacks. Practice to score 1500+ marks<\/p>\n","protected":false},"author":1,"featured_media":8631,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"googlesitekit_rrm_CAowmvTFDA:productID":"","_coblocks_attr":"","_coblocks_dimensions":"","_coblocks_responsive_height":"","_coblocks_accordion_ie_support":"","footnotes":""},"categories":[13,18,4],"tags":[26,27,29],"class_list":["post-8338","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-2nd-module","category-sat-2026","category-sat-math","tag-sat-2026","tag-sat-math","tag-sat-module-2nd"],"aioseo_notices":[],"_links":{"self":[{"href":"https:\/\/us.mrenglishkj.com\/sat\/wp-json\/wp\/v2\/posts\/8338","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/us.mrenglishkj.com\/sat\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/us.mrenglishkj.com\/sat\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/us.mrenglishkj.com\/sat\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/us.mrenglishkj.com\/sat\/wp-json\/wp\/v2\/comments?post=8338"}],"version-history":[{"count":3,"href":"https:\/\/us.mrenglishkj.com\/sat\/wp-json\/wp\/v2\/posts\/8338\/revisions"}],"predecessor-version":[{"id":8902,"href":"https:\/\/us.mrenglishkj.com\/sat\/wp-json\/wp\/v2\/posts\/8338\/revisions\/8902"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/us.mrenglishkj.com\/sat\/wp-json\/wp\/v2\/media\/8631"}],"wp:attachment":[{"href":"https:\/\/us.mrenglishkj.com\/sat\/wp-json\/wp\/v2\/media?parent=8338"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/us.mrenglishkj.com\/sat\/wp-json\/wp\/v2\/categories?post=8338"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/us.mrenglishkj.com\/sat\/wp-json\/wp\/v2\/tags?post=8338"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}