{"id":1290,"date":"2024-05-08T01:16:15","date_gmt":"2024-05-08T01:16:15","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/?post_type=chapter&#038;p=1290"},"modified":"2025-01-16T21:21:58","modified_gmt":"2025-01-16T21:21:58","slug":"quadratic-equations-fresh-take","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/chapter\/quadratic-equations-fresh-take\/","title":{"raw":"Quadratic Equations: Fresh Take","rendered":"Quadratic Equations: Fresh Take"},"content":{"raw":"<section class=\"textbox learningGoals\">\r\n<ul>\r\n \t<li>Solve quadratic equations by factoring.<\/li>\r\n \t<li>Solve quadratic equations by square root property.<\/li>\r\n \t<li>Solve quadratic equations by completing the square.<\/li>\r\n \t<li>Solve quadratic equations by using quadratic formula.<\/li>\r\n<\/ul>\r\n<\/section>\r\n<h2>Solving Quadratic Equations by Factoring<\/h2>\r\n<div class=\"textbox shaded\">\r\n\r\n<strong>The Main Idea\r\n<\/strong>\r\n<ul>\r\n \t<li class=\"whitespace-normal break-words\">Definition of Quadratic Equations:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Second-degree polynomial equation<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Standard form: [latex]ax^2 + bx + c = 0[\/latex], where [latex]a \\neq 0[\/latex]<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Zero-Product Property:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">If [latex]ab = 0[\/latex], then [latex]a = 0[\/latex] or [latex]b = 0[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Fundamental to solving factored quadratics<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li>Grouping Method:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Used when the leading coefficient is not 1<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Transforms the quadratic into four terms for easier factoring<\/li>\r\n \t<li>Key Steps:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Find factors of [latex]ac[\/latex] that sum to [latex]b[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Rewrite middle term using these factors<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Group terms and factor by pairs<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Factor out common binomial<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ul>\r\n<p class=\"font-600 text-xl font-bold\"><strong>Solving by Factoring: Step-by-Step<\/strong><\/p>\r\n\r\n<ol class=\"-mt-1 list-decimal space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Arrange the equation in standard form: [latex]ax^2 + bx + c = 0[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Factor the left side of the equation<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Apply the zero-product property<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Solve the resulting linear equations<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Check solutions in the original equation<\/li>\r\n<\/ol>\r\n<\/div>\r\n<section class=\"textbox example\" aria-label=\"Example\">Factor and solve the quadratic equation: [latex]{x}^{2}-5x - 6=0[\/latex].[reveal-answer q=\"220537\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"220537\"][latex]\\left(x - 6\\right)\\left(x+1\\right)=0;x=6,x=-1[\/latex][\/hidden-answer]<\/section><section class=\"textbox watchIt\" aria-label=\"Watch It\">Watch the following video for more examples of factoring quadratics with a leading coefficient of 1.\r\n<script type=\"text\/javascript\" src=\"https:\/\/www.youtube.com\/iframe_api \"><\/script>\r\n<p class=\"cc-media-iframe-container\"><iframe id=\"tpm-plugin-ecbggdfh-eF6zYNzlZKQ\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/eF6zYNzlZKQ?enablejsapi=1 \" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\r\n\r\n<div id=\"3p-plugin-target-ecbggdfh-eF6zYNzlZKQ\" class=\"p3sdk-target\"><\/div>\r\n<p class=\"cc-media-iframe-container\"><script type='text\/javascript' src='\/\/plugin.3playmedia.com\/ajax.js?cc=1&cc_minimizable=1&cc_minimize_on_load=0&cc_multi_text_track=0&cc_overlay=1&cc_searchable=0&embed=ajax&mf=12844236&p3sdk_version=1.11.7&p=20361&player_type=youtube&plugin_skin=dark&target=3p-plugin-target-ecbggdfh-eF6zYNzlZKQ&vembed=0&video_id=eF6zYNzlZKQ&video_target=tpm-plugin-ecbggdfh-eF6zYNzlZKQ'><\/script><\/p>\r\nYou can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/College+Algebra+Corequisite\/Transcripts\/More+examples+of+factoring+quadratics+with+a+leading+coefficient+of+1+%7C+Algebra+II+%7C+Khan+Academy_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cMore examples of factoring quadratics with a leading coefficient of 1 | Algebra II | Khan Academy\u201d here (opens in new window).<\/a>\r\n\r\n<\/section><section class=\"textbox example\" aria-label=\"Example\">Factor and solve the equation: [latex]2x^2+7x =-3[\/latex].[reveal-answer q=\"486286\"]<span style=\"font-family: 'Public Sans', -apple-system, BlinkMacSystemFont, 'Segoe UI', Roboto, Oxygen-Sans, Ubuntu, Cantarell, 'Helvetica Neue', sans-serif;\">Show Answer[\/reveal-answer]<\/span><span style=\"font-family: 'Public Sans', -apple-system, BlinkMacSystemFont, 'Segoe UI', Roboto, Oxygen-Sans, Ubuntu, Cantarell, 'Helvetica Neue', sans-serif;\">[hidden-answer a=\"486286\"]<\/span>First, we should re-write it to the standard form: [latex]2x^2+7x +3=0[\/latex]\r\n<ul>\r\n \t<li>We have a trinomial with coefficients [latex]a = 2[\/latex], [latex]b = 7[\/latex], and [latex]c = 3[\/latex].<\/li>\r\n \t<li>We need to find two numbers with a product of [latex]ac = 2 \\cdot 3 = 6[\/latex] and a sum of [latex]7[\/latex].<\/li>\r\n<\/ul>\r\n<table style=\"border-collapse: collapse; width: 100%;\">\r\n<tbody>\r\n<tr>\r\n<td style=\"width: 50%;\"><strong>Factors of [latex]-6[\/latex]<\/strong><\/td>\r\n<td style=\"width: 50%;\"><strong>Sum of Factors<\/strong><\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 50%;\">[latex]1, 6[\/latex]<\/td>\r\n<td style=\"width: 50%;\">[latex]7[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 50%;\">[latex]2, 3[\/latex]<\/td>\r\n<td style=\"width: 50%;\">[latex]5[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n\\begin{align*} \\text{Equation:} &amp; \\quad 2x^2 + 7x + 3 = 0 \\\\ \\text{Find numbers that multiply to \\(6\\) and add to \\(7\\):} &amp; \\quad 6, 1 \\\\ \\text{Rewrite the equation using these numbers:} &amp; \\quad 2x^2 + 6x + 1x + 3 = 0 \\\\ \\text{Group terms and factor:} &amp; \\\\ &amp; \\quad (2x^2 + 6x) + (1x + 3) = 0 \\\\ &amp; \\quad 2x(x + 3) + 1(x + 3) = 0 \\\\ \\text{Factor out the common binomial factor:} &amp; \\\\ &amp; \\quad (2x + 1)(x + 3) = 0 \\\\ \\text{Apply the zero product property:} &amp; \\\\ &amp; \\quad 2x + 1 = 0 \\quad \\text{or} \\quad x + 3 = 0 \\\\ \\text{Solve each equation:} &amp; \\\\ &amp; \\quad 2x + 1 = 0 \\Rightarrow 2x = -1 \\Rightarrow x = -\\frac{1}{2} \\\\ &amp; \\quad x + 3 = 0 \\Rightarrow x = -3 \\end{align*}\r\n\r\n[\/hidden-answer]\r\n\r\n<\/section>\r\n<h2>Using the Square Root Property<\/h2>\r\n<div class=\"textbox shaded\">\r\n\r\n<strong>The Main Idea\r\n<\/strong>\r\n<ul>\r\n \t<li class=\"whitespace-normal break-words\">Square Root Property:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Used when there's no linear term in the quadratic equation<\/li>\r\n \t<li class=\"whitespace-normal break-words\">If [latex]x^2 = k[\/latex], then [latex]x = \\pm\\sqrt{k}[\/latex] (where [latex]k \\geq 0[\/latex])<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Applying Square Root Property:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Isolate [latex]x^2[\/latex] term<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Take square root of both sides<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Remember to use [latex]\\pm[\/latex] sign<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Pythagorean Theorem:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Relates sides of a right triangle: [latex]a^2 + b^2 = c^2[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">[latex]a[\/latex] and [latex]b[\/latex] are legs, [latex]c[\/latex] is hypotenuse<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Leads to quadratic equations when solving for a side<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ul>\r\n<\/div>\r\n<section class=\"textbox example\" aria-label=\"Example\">Solve the quadratic equation: [latex]4{x}^{2}+1=7[\/latex][reveal-answer q=\"885054\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"885054\"]First, isolate the [latex]{x}^{2}[\/latex] term. Then take the square root of both sides.\r\n<div style=\"text-align: center;\">[latex]\\begin{array}{l}4{x}^{2}+1=7\\hfill \\\\ 4{x}^{2}=6\\hfill \\\\ {x}^{2}=\\frac{6}{4}\\hfill \\\\ x=\\pm \\frac{\\sqrt{6}}{2}\\hfill \\end{array}[\/latex]<\/div>\r\nThe solutions are [latex]x=\\frac{\\sqrt{6}}{2}[\/latex], [latex]x=-\\frac{\\sqrt{6}}{2}[\/latex].\r\n\r\n[\/hidden-answer]\r\n\r\n<\/section><section class=\"textbox watchIt\" aria-label=\"Watch It\"><script type=\"text\/javascript\" src=\"https:\/\/www.youtube.com\/iframe_api \"><\/script>\r\n<p class=\"cc-media-iframe-container\"><iframe id=\"tpm-plugin-ccebcefe-RweAgQwLdMs\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/RweAgQwLdMs?enablejsapi=1 \" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\r\n\r\n<div id=\"3p-plugin-target-ccebcefe-RweAgQwLdMs\" class=\"p3sdk-target\"><\/div>\r\n<p class=\"cc-media-iframe-container\"><script type='text\/javascript' src='\/\/plugin.3playmedia.com\/ajax.js?cc=1&cc_minimizable=1&cc_minimize_on_load=0&cc_multi_text_track=0&cc_overlay=1&cc_searchable=0&embed=ajax&mf=12844257&p3sdk_version=1.11.7&p=20361&player_type=youtube&plugin_skin=dark&target=3p-plugin-target-ccebcefe-RweAgQwLdMs&vembed=0&video_id=RweAgQwLdMs&video_target=tpm-plugin-ccebcefe-RweAgQwLdMs'><\/script><\/p>\r\nYou can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/College+Algebra+Corequisite\/Transcripts\/Example+-+Solving+simple+quadratic+%7C+Quadratic+equations+%7C+Algebra+I+%7C+Khan+Academy_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cExample: Solving simple quadratic | Quadratic equations | Algebra I | Khan Academy\u201d here (opens in new window).<\/a>\r\n\r\n<\/section>\r\n<h2>Completing the Square<\/h2>\r\n<div class=\"textbox shaded\">\r\n\r\n<strong>The Main Idea\r\n<\/strong>\r\n<ul>\r\n \t<li class=\"whitespace-normal break-words\">Purpose:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Transform [latex]ax^2 + bx + c = 0[\/latex] into [latex](x - h)^2 = k[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Enables solving quadratics that can't be easily factored<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Key Steps:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li>Rearrange the equation so that it is in standard form<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Isolate [latex]x[\/latex] terms on one side<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Add [latex](\\frac{b}{2})^2[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Factor perfect square trinomial<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Apply square root property<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Perfect Square Trinomials:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">[latex]x^2 + 2px + p^2 = (x + p)^2[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">[latex]x^2 - 2px + p^2 = (x - p)^2[\/latex]<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Applications:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Solving quadratic equations<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Finding vertex of parabolas<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Deriving quadratic formula<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ul>\r\n<strong>\u00a0<\/strong>\r\n\r\n<\/div>\r\n<section class=\"textbox example\" aria-label=\"Example\">Solve by completing the square: [latex]{x}^{2}-6x=13[\/latex].[reveal-answer q=\"222291\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"222291\"][latex]x=3\\pm \\sqrt{22}[\/latex][\/hidden-answer]<\/section><section class=\"textbox watchIt\" aria-label=\"Watch It\"><script type=\"text\/javascript\" src=\"https:\/\/www.youtube.com\/iframe_api \"><\/script>\r\n<p class=\"cc-media-iframe-container\"><iframe id=\"tpm-plugin-begfbefg-bNQY0z76M5A\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/bNQY0z76M5A?enablejsapi=1 \" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\r\n\r\n<div id=\"3p-plugin-target-begfbefg-bNQY0z76M5A\" class=\"p3sdk-target\"><\/div>\r\n<p class=\"cc-media-iframe-container\"><script type='text\/javascript' src='\/\/plugin.3playmedia.com\/ajax.js?cc=1&cc_minimizable=1&cc_minimize_on_load=0&cc_multi_text_track=0&cc_overlay=1&cc_searchable=0&embed=ajax&mf=12844258&p3sdk_version=1.11.7&p=20361&player_type=youtube&plugin_skin=dark&target=3p-plugin-target-begfbefg-bNQY0z76M5A&vembed=0&video_id=bNQY0z76M5A&video_target=tpm-plugin-begfbefg-bNQY0z76M5A'><\/script><\/p>\r\nYou can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/College+Algebra+Corequisite\/Transcripts\/Solving+quadratic+equations+by+completing+the+square+%7C+Algebra+II+%7C+Khan+Academy_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cSolving quadratic equations by completing the square | Algebra II | Khan Academy\u201d here (opens in new window).<\/a>\r\n\r\n<\/section>\r\n<h2>Quadratic Formula<\/h2>\r\n<div class=\"textbox shaded\">\r\n\r\n<strong>The Main Idea\r\n<\/strong>\r\n<ul>\r\n \t<li class=\"whitespace-normal break-words\">Quadratic Formula:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Solves any quadratic equation [latex]ax^2 + bx + c = 0[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Formula: [latex]x = \\frac{-b \\pm \\sqrt{b^2 - 4ac}}{2a}[\/latex]<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ul>\r\n<\/div>\r\n<section class=\"textbox example\">Solve the quadratic equation using the quadratic formula: [latex]9{x}^{2}+3x - 2=0[\/latex].[reveal-answer q=\"232269\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"232269\"][latex]x=-\\frac{2}{3}[\/latex], [latex]x=\\frac{1}{3}[\/latex][\/hidden-answer]<\/section>","rendered":"<section class=\"textbox learningGoals\">\n<ul>\n<li>Solve quadratic equations by factoring.<\/li>\n<li>Solve quadratic equations by square root property.<\/li>\n<li>Solve quadratic equations by completing the square.<\/li>\n<li>Solve quadratic equations by using quadratic formula.<\/li>\n<\/ul>\n<\/section>\n<h2>Solving Quadratic Equations by Factoring<\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea<br \/>\n<\/strong><\/p>\n<ul>\n<li class=\"whitespace-normal break-words\">Definition of Quadratic Equations:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Second-degree polynomial equation<\/li>\n<li class=\"whitespace-normal break-words\">Standard form: [latex]ax^2 + bx + c = 0[\/latex], where [latex]a \\neq 0[\/latex]<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Zero-Product Property:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">If [latex]ab = 0[\/latex], then [latex]a = 0[\/latex] or [latex]b = 0[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Fundamental to solving factored quadratics<\/li>\n<\/ul>\n<\/li>\n<li>Grouping Method:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Used when the leading coefficient is not 1<\/li>\n<li class=\"whitespace-normal break-words\">Transforms the quadratic into four terms for easier factoring<\/li>\n<li>Key Steps:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Find factors of [latex]ac[\/latex] that sum to [latex]b[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Rewrite middle term using these factors<\/li>\n<li class=\"whitespace-normal break-words\">Group terms and factor by pairs<\/li>\n<li class=\"whitespace-normal break-words\">Factor out common binomial<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<p class=\"font-600 text-xl font-bold\"><strong>Solving by Factoring: Step-by-Step<\/strong><\/p>\n<ol class=\"-mt-1 list-decimal space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Arrange the equation in standard form: [latex]ax^2 + bx + c = 0[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Factor the left side of the equation<\/li>\n<li class=\"whitespace-normal break-words\">Apply the zero-product property<\/li>\n<li class=\"whitespace-normal break-words\">Solve the resulting linear equations<\/li>\n<li class=\"whitespace-normal break-words\">Check solutions in the original equation<\/li>\n<\/ol>\n<\/div>\n<section class=\"textbox example\" aria-label=\"Example\">Factor and solve the quadratic equation: [latex]{x}^{2}-5x - 6=0[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q220537\">Show Solution<\/button><\/p>\n<div id=\"q220537\" class=\"hidden-answer\" style=\"display: none\">[latex]\\left(x - 6\\right)\\left(x+1\\right)=0;x=6,x=-1[\/latex]<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox watchIt\" aria-label=\"Watch It\">Watch the following video for more examples of factoring quadratics with a leading coefficient of 1.<br \/>\n<script type=\"text\/javascript\" src=\"https:\/\/www.youtube.com\/iframe_api\"><\/script><\/p>\n<p class=\"cc-media-iframe-container\"><iframe id=\"tpm-plugin-ecbggdfh-eF6zYNzlZKQ\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/eF6zYNzlZKQ?enablejsapi=1\" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\n<div id=\"3p-plugin-target-ecbggdfh-eF6zYNzlZKQ\" class=\"p3sdk-target\"><\/div>\n<p class=\"cc-media-iframe-container\"><script type=\"text\/javascript\" src=\"\/\/plugin.3playmedia.com\/ajax.js?cc=1&#38;cc_minimizable=1&#38;cc_minimize_on_load=0&#38;cc_multi_text_track=0&#38;cc_overlay=1&#38;cc_searchable=0&#38;embed=ajax&#38;mf=12844236&#38;p3sdk_version=1.11.7&#38;p=20361&#38;player_type=youtube&#38;plugin_skin=dark&#38;target=3p-plugin-target-ecbggdfh-eF6zYNzlZKQ&#38;vembed=0&#38;video_id=eF6zYNzlZKQ&#38;video_target=tpm-plugin-ecbggdfh-eF6zYNzlZKQ\"><\/script><\/p>\n<p>You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/College+Algebra+Corequisite\/Transcripts\/More+examples+of+factoring+quadratics+with+a+leading+coefficient+of+1+%7C+Algebra+II+%7C+Khan+Academy_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cMore examples of factoring quadratics with a leading coefficient of 1 | Algebra II | Khan Academy\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n<section class=\"textbox example\" aria-label=\"Example\">Factor and solve the equation: [latex]2x^2+7x =-3[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q486286\"><span style=\"font-family: 'Public Sans', -apple-system, BlinkMacSystemFont, 'Segoe UI', Roboto, Oxygen-Sans, Ubuntu, Cantarell, 'Helvetica Neue', sans-serif;\">Show Answer<\/button><\/span><span style=\"font-family: 'Public Sans', -apple-system, BlinkMacSystemFont, 'Segoe UI', Roboto, Oxygen-Sans, Ubuntu, Cantarell, 'Helvetica Neue', sans-serif;\"><\/p>\n<div id=\"q486286\" class=\"hidden-answer\" style=\"display: none\"><\/span>First, we should re-write it to the standard form: [latex]2x^2+7x +3=0[\/latex]<\/p>\n<ul>\n<li>We have a trinomial with coefficients [latex]a = 2[\/latex], [latex]b = 7[\/latex], and [latex]c = 3[\/latex].<\/li>\n<li>We need to find two numbers with a product of [latex]ac = 2 \\cdot 3 = 6[\/latex] and a sum of [latex]7[\/latex].<\/li>\n<\/ul>\n<table style=\"border-collapse: collapse; width: 100%;\">\n<tbody>\n<tr>\n<td style=\"width: 50%;\"><strong>Factors of [latex]-6[\/latex]<\/strong><\/td>\n<td style=\"width: 50%;\"><strong>Sum of Factors<\/strong><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 50%;\">[latex]1, 6[\/latex]<\/td>\n<td style=\"width: 50%;\">[latex]7[\/latex]<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 50%;\">[latex]2, 3[\/latex]<\/td>\n<td style=\"width: 50%;\">[latex]5[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>\\begin{align*} \\text{Equation:} &amp; \\quad 2x^2 + 7x + 3 = 0 \\\\ \\text{Find numbers that multiply to \\(6\\) and add to \\(7\\):} &amp; \\quad 6, 1 \\\\ \\text{Rewrite the equation using these numbers:} &amp; \\quad 2x^2 + 6x + 1x + 3 = 0 \\\\ \\text{Group terms and factor:} &amp; \\\\ &amp; \\quad (2x^2 + 6x) + (1x + 3) = 0 \\\\ &amp; \\quad 2x(x + 3) + 1(x + 3) = 0 \\\\ \\text{Factor out the common binomial factor:} &amp; \\\\ &amp; \\quad (2x + 1)(x + 3) = 0 \\\\ \\text{Apply the zero product property:} &amp; \\\\ &amp; \\quad 2x + 1 = 0 \\quad \\text{or} \\quad x + 3 = 0 \\\\ \\text{Solve each equation:} &amp; \\\\ &amp; \\quad 2x + 1 = 0 \\Rightarrow 2x = -1 \\Rightarrow x = -\\frac{1}{2} \\\\ &amp; \\quad x + 3 = 0 \\Rightarrow x = -3 \\end{align*}<\/p>\n<\/div>\n<\/div>\n<\/section>\n<h2>Using the Square Root Property<\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea<br \/>\n<\/strong><\/p>\n<ul>\n<li class=\"whitespace-normal break-words\">Square Root Property:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Used when there&#8217;s no linear term in the quadratic equation<\/li>\n<li class=\"whitespace-normal break-words\">If [latex]x^2 = k[\/latex], then [latex]x = \\pm\\sqrt{k}[\/latex] (where [latex]k \\geq 0[\/latex])<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Applying Square Root Property:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Isolate [latex]x^2[\/latex] term<\/li>\n<li class=\"whitespace-normal break-words\">Take square root of both sides<\/li>\n<li class=\"whitespace-normal break-words\">Remember to use [latex]\\pm[\/latex] sign<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Pythagorean Theorem:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Relates sides of a right triangle: [latex]a^2 + b^2 = c^2[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">[latex]a[\/latex] and [latex]b[\/latex] are legs, [latex]c[\/latex] is hypotenuse<\/li>\n<li class=\"whitespace-normal break-words\">Leads to quadratic equations when solving for a side<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<\/div>\n<section class=\"textbox example\" aria-label=\"Example\">Solve the quadratic equation: [latex]4{x}^{2}+1=7[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q885054\">Show Solution<\/button><\/p>\n<div id=\"q885054\" class=\"hidden-answer\" style=\"display: none\">First, isolate the [latex]{x}^{2}[\/latex] term. Then take the square root of both sides.<\/p>\n<div style=\"text-align: center;\">[latex]\\begin{array}{l}4{x}^{2}+1=7\\hfill \\\\ 4{x}^{2}=6\\hfill \\\\ {x}^{2}=\\frac{6}{4}\\hfill \\\\ x=\\pm \\frac{\\sqrt{6}}{2}\\hfill \\end{array}[\/latex]<\/div>\n<p>The solutions are [latex]x=\\frac{\\sqrt{6}}{2}[\/latex], [latex]x=-\\frac{\\sqrt{6}}{2}[\/latex].<\/p>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox watchIt\" aria-label=\"Watch It\"><script type=\"text\/javascript\" src=\"https:\/\/www.youtube.com\/iframe_api\"><\/script><\/p>\n<p class=\"cc-media-iframe-container\"><iframe id=\"tpm-plugin-ccebcefe-RweAgQwLdMs\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/RweAgQwLdMs?enablejsapi=1\" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\n<div id=\"3p-plugin-target-ccebcefe-RweAgQwLdMs\" class=\"p3sdk-target\"><\/div>\n<p class=\"cc-media-iframe-container\"><script type=\"text\/javascript\" src=\"\/\/plugin.3playmedia.com\/ajax.js?cc=1&#38;cc_minimizable=1&#38;cc_minimize_on_load=0&#38;cc_multi_text_track=0&#38;cc_overlay=1&#38;cc_searchable=0&#38;embed=ajax&#38;mf=12844257&#38;p3sdk_version=1.11.7&#38;p=20361&#38;player_type=youtube&#38;plugin_skin=dark&#38;target=3p-plugin-target-ccebcefe-RweAgQwLdMs&#38;vembed=0&#38;video_id=RweAgQwLdMs&#38;video_target=tpm-plugin-ccebcefe-RweAgQwLdMs\"><\/script><\/p>\n<p>You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/College+Algebra+Corequisite\/Transcripts\/Example+-+Solving+simple+quadratic+%7C+Quadratic+equations+%7C+Algebra+I+%7C+Khan+Academy_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cExample: Solving simple quadratic | Quadratic equations | Algebra I | Khan Academy\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n<h2>Completing the Square<\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea<br \/>\n<\/strong><\/p>\n<ul>\n<li class=\"whitespace-normal break-words\">Purpose:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Transform [latex]ax^2 + bx + c = 0[\/latex] into [latex](x - h)^2 = k[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Enables solving quadratics that can&#8217;t be easily factored<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Key Steps:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li>Rearrange the equation so that it is in standard form<\/li>\n<li class=\"whitespace-normal break-words\">Isolate [latex]x[\/latex] terms on one side<\/li>\n<li class=\"whitespace-normal break-words\">Add [latex](\\frac{b}{2})^2[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Factor perfect square trinomial<\/li>\n<li class=\"whitespace-normal break-words\">Apply square root property<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Perfect Square Trinomials:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">[latex]x^2 + 2px + p^2 = (x + p)^2[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">[latex]x^2 - 2px + p^2 = (x - p)^2[\/latex]<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Applications:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Solving quadratic equations<\/li>\n<li class=\"whitespace-normal break-words\">Finding vertex of parabolas<\/li>\n<li class=\"whitespace-normal break-words\">Deriving quadratic formula<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<p><strong>\u00a0<\/strong><\/p>\n<\/div>\n<section class=\"textbox example\" aria-label=\"Example\">Solve by completing the square: [latex]{x}^{2}-6x=13[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q222291\">Show Solution<\/button><\/p>\n<div id=\"q222291\" class=\"hidden-answer\" style=\"display: none\">[latex]x=3\\pm \\sqrt{22}[\/latex]<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox watchIt\" aria-label=\"Watch It\"><script type=\"text\/javascript\" src=\"https:\/\/www.youtube.com\/iframe_api\"><\/script><\/p>\n<p class=\"cc-media-iframe-container\"><iframe id=\"tpm-plugin-begfbefg-bNQY0z76M5A\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/bNQY0z76M5A?enablejsapi=1\" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\n<div id=\"3p-plugin-target-begfbefg-bNQY0z76M5A\" class=\"p3sdk-target\"><\/div>\n<p class=\"cc-media-iframe-container\"><script type=\"text\/javascript\" src=\"\/\/plugin.3playmedia.com\/ajax.js?cc=1&#38;cc_minimizable=1&#38;cc_minimize_on_load=0&#38;cc_multi_text_track=0&#38;cc_overlay=1&#38;cc_searchable=0&#38;embed=ajax&#38;mf=12844258&#38;p3sdk_version=1.11.7&#38;p=20361&#38;player_type=youtube&#38;plugin_skin=dark&#38;target=3p-plugin-target-begfbefg-bNQY0z76M5A&#38;vembed=0&#38;video_id=bNQY0z76M5A&#38;video_target=tpm-plugin-begfbefg-bNQY0z76M5A\"><\/script><\/p>\n<p>You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/College+Algebra+Corequisite\/Transcripts\/Solving+quadratic+equations+by+completing+the+square+%7C+Algebra+II+%7C+Khan+Academy_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cSolving quadratic equations by completing the square | Algebra II | Khan Academy\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n<h2>Quadratic Formula<\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea<br \/>\n<\/strong><\/p>\n<ul>\n<li class=\"whitespace-normal break-words\">Quadratic Formula:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Solves any quadratic equation [latex]ax^2 + bx + c = 0[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Formula: [latex]x = \\frac{-b \\pm \\sqrt{b^2 - 4ac}}{2a}[\/latex]<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<\/div>\n<section class=\"textbox example\">Solve the quadratic equation using the quadratic formula: [latex]9{x}^{2}+3x - 2=0[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q232269\">Show Solution<\/button><\/p>\n<div id=\"q232269\" class=\"hidden-answer\" style=\"display: none\">[latex]x=-\\frac{2}{3}[\/latex], [latex]x=\\frac{1}{3}[\/latex]<\/div>\n<\/div>\n<\/section>\n","protected":false},"author":12,"menu_order":10,"template":"","meta":{"_candela_citation":"[{\"type\":\"copyrighted_video\",\"description\":\"More examples of factoring quadratics with a 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