{"id":15390,"date":"2021-10-11T23:05:42","date_gmt":"2021-10-11T23:05:42","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/ccbcmd-math-1\/chapter\/read-the-discriminant\/"},"modified":"2021-10-18T03:45:55","modified_gmt":"2021-10-18T03:45:55","slug":"the-discriminant","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/ccbcmd-math-1\/chapter\/the-discriminant\/","title":{"raw":"The Discriminant","rendered":"The Discriminant"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h3>Learning Outcome<\/h3>\r\n<ul>\r\n \t<li>Use the discriminant to classify solutions to quadratic equations<\/li>\r\n<\/ul>\r\n<\/div>\r\n<h2>The Discriminant<\/h2>\r\nThe <strong>quadratic formula<\/strong> not only generates the solutions to a quadratic equation, but can also tell us about the nature of the solutions. When we consider the <strong>discriminant<\/strong>, or the expression under the radical, [latex]{b}^{2}-4ac[\/latex], it tells us whether the solutions are real numbers or complex numbers and how many solutions of each type to expect.\r\n\r\nLet us explore how the discriminant affects the evaluation of [latex] \\sqrt{{{b}^{2}}-4ac}[\/latex] in the quadratic formula and how it helps to determine the solution set.\r\n<ul>\r\n \t<li>If [latex]b^{2}-4ac&gt;0[\/latex], then the number underneath the radical will be a positive value. You can always find the square root of a positive number, so evaluating the quadratic formula will result in two real solutions (one by adding the positive square root and one by subtracting it).<\/li>\r\n \t<li>If [latex]b^{2}-4ac=0[\/latex], then you will be taking the square root of\u00a0[latex]0[\/latex], which is\u00a0[latex]0[\/latex]. Since adding and subtracting\u00a0[latex]0[\/latex] both give the same result, the \"[latex]\\pm[\/latex]\" portion of the formula does not matter. There will be one real repeated solution.<\/li>\r\n \t<li>If [latex]b^{2}-4ac&lt;0[\/latex], then the number underneath the radical will be a negative value. Since you cannot find the square root of a negative number using real numbers, there are no real solutions. However, you can use imaginary numbers. You will then have two complex solutions, one by adding the imaginary square root and one by subtracting it.<\/li>\r\n<\/ul>\r\nThe table below summarizes the relationship between the value of the discriminant and the solutions of a quadratic equation.\r\n<table style=\"width: 60%;\" summary=\"A table with 5 rows and 2 columns. The entries in the first row are: Value of Discriminant and Results. The entries in the second row are: b squared minus four times a times c equals zero and One rational solution (double solution). The entries in the third row are: b squared minus four times a times c is greater than zero, perfect square and Two rational solutions. The entries in the fourth row are: b squared minus four times a times c is greater than zero, not a perfect square and Two irrational solutions. The entries in the fifth row are: b squared minus four times a times c is less than zero and Two complex solutions.\">\r\n<thead>\r\n<tr>\r\n<th>Value of Discriminant<\/th>\r\n<th>Results<\/th>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr>\r\n<td>[latex]{b}^{2}-4ac=0[\/latex]<\/td>\r\n<td>One repeated rational solution<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]{b}^{2}-4ac&gt;0[\/latex], perfect square<\/td>\r\n<td>Two rational solutions<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]{b}^{2}-4ac&gt;0[\/latex], not a perfect square<\/td>\r\n<td>Two irrational solutions<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]{b}^{2}-4ac&lt;0[\/latex]<\/td>\r\n<td>Two complex solutions<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<div class=\"shaded textbox\">\r\n<h3>The Discriminant<\/h3>\r\nFor [latex]a{x}^{2}+bx+c=0[\/latex], where [latex]a[\/latex], [latex]b[\/latex], and [latex]c[\/latex] are real numbers, the <strong>discriminant<\/strong> is the expression under the radical in the quadratic formula: [latex]{b}^{2}-4ac[\/latex]. It tells us whether the solutions are real numbers or complex numbers and how many solutions of each type to expect.\r\n\r\n<\/div>\r\n<div class=\"bcc-box bcc-info\">\r\n<h3>Example<\/h3>\r\nUse the discriminant to determine how many and what kind of solutions the quadratic equation [latex]x^{2}-4x+10=0[\/latex]\u00a0has.\r\n\r\n[reveal-answer q=\"116245\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"116245\"]\r\n\r\nEvaluate [latex]b^{2}-4ac[\/latex]. First note that\u00a0[latex]a=1,b=\u22124[\/latex], and [latex]c=10[\/latex].\r\n\r\n[latex]\\begin{array}{l}b^{2}-4ac=\\left(-4\\right)^{2}-4\\left(1\\right)\\left(10\\right)=16-40=-24\\end{array}[\/latex]\r\n\r\nThe result is a negative number. The discriminant is negative, so the quadratic equation has two complex solutions.\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nUse the discriminant to find the nature of the solutions to the following quadratic equations:\r\n<ol>\r\n \t<li>[latex]{x}^{2}+4x+4=0[\/latex]<\/li>\r\n \t<li>[latex]8{x}^{2}+14x+3=0[\/latex]<\/li>\r\n \t<li>[latex]3{x}^{2}-5x - 2=0[\/latex]<\/li>\r\n \t<li>[latex]3{x}^{2}-10x+15=0[\/latex]<\/li>\r\n<\/ol>\r\n[reveal-answer q=\"497176\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"497176\"]\r\n\r\nCalculate the discriminant [latex]{b}^{2}-4ac[\/latex] for each equation and state the expected type of solutions.\r\n<ol>\r\n \t<li style=\"text-align: left;\">[latex]{x}^{2}+4x+4=0[\/latex] [latex] \\\\ {b}^{2}-4ac={\\left(4\\right)}^{2}-4\\left(1\\right)\\left(4\\right)=0[\/latex] [latex]\\text{There will be one repeated rational solution.}[\/latex]<\/li>\r\n \t<li style=\"text-align: left;\">[latex]8{x}^{2}+14x+3=0[\/latex][latex] \\\\ {b}^{2}-4ac={\\left(14\\right)}^{2}-4\\left(8\\right)\\left(3\\right)=100[\/latex] [latex]\\text{100 is a perfect square, so there will be two rational solutions.}[\/latex]<\/li>\r\n \t<li style=\"text-align: left;\">[latex]3{x}^{2}-5x - 2=0[\/latex][latex] \\\\ {b}^{2}-4ac={\\left(-5\\right)}^{2}-4\\left(3\\right)\\left(-2\\right)=49[\/latex] [latex]\\text{49 is a perfect square, so there will be two rational solutions.}[\/latex]<\/li>\r\n \t<li style=\"text-align: left;\">[latex]3{x}^{2}-10x+15=0[\/latex][latex] \\\\ {b}^{2}-4ac={\\left(-10\\right)}^{2}-4\\left(3\\right)\\left(15\\right)=-80[\/latex] [latex]\\text{There will be two complex solutions.}[\/latex]<\/li>\r\n<\/ol>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\n[ohm_question]2977[\/ohm_question]\r\n\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n\r\n&nbsp;\r\n\r\n<\/div>\r\nWe can summarize our results as follows:\r\n<table>\r\n<tbody>\r\n<tr>\r\n<td>Discriminant<\/td>\r\n<td>Number and Type of Solutions<\/td>\r\n<td>Graph of\u00a0[latex]y=ax^2+bx+c[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]b^{2}-4ac&lt;0[\/latex]<\/td>\r\n<td>two complex solutions<\/td>\r\n<td>will not cross the x-axis<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]b^{2}-4ac=0[\/latex]<\/td>\r\n<td>one real repeated solution<\/td>\r\n<td>will touch x-axis once<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]b^{2}-4ac&gt;0[\/latex]<\/td>\r\n<td>\u00a0two real solutions<\/td>\r\n<td>\u00a0will cross x-axis twice<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nIn the following video, we show more examples of how to use the discriminant to describe the type of solutions of a quadratic equation.\r\n\r\nhttps:\/\/youtu.be\/hSWs0VUyn1k\r\n<h2>Summary<\/h2>\r\nThe discriminant of the quadratic formula is the quantity under the radical, [latex] {{b}^{2}}-4ac[\/latex]. It determines the number and the type of solutions that a quadratic equation has. If the discriminant is positive, there are\u00a0[latex]2[\/latex] real solutions. If it is\u00a0[latex]0[\/latex], there is\u00a0[latex]1[\/latex] real repeated solution. If the discriminant is negative, there are\u00a0[latex]2[\/latex] complex solutions (but no real solutions).\r\n\r\nThe discriminant can also tell us about the behavior of the graph of a quadratic equation in two variables.\r\n<h2>Contribute!<\/h2>\r\n<div style=\"margin-bottom: 8px;\">Did you have an idea for improving this content? We\u2019d love your input.<\/div>\r\n<a style=\"font-size: 10pt; font-weight: 600; color: #077fab; text-decoration: none; border: 2px solid #077fab; border-radius: 7px; padding: 5px 25px; text-align: center; cursor: pointer; line-height: 1.5em;\" href=\"https:\/\/docs.google.com\/document\/d\/12KLcUeXJOwUh7viWyV3U6UXSmyJAEKrzYcQuzHXYuDU\" target=\"_blank\" rel=\"noopener\">Improve this page<\/a><a style=\"margin-left: 16px;\" href=\"https:\/\/docs.google.com\/document\/d\/1vy-T6DtTF-BbMfpVEI7VP_R7w2A4anzYZLXR8Pk4Fu4\" target=\"_blank\" rel=\"noopener\">Learn More<\/a>","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Learning Outcome<\/h3>\n<ul>\n<li>Use the discriminant to classify solutions to quadratic equations<\/li>\n<\/ul>\n<\/div>\n<h2>The Discriminant<\/h2>\n<p>The <strong>quadratic formula<\/strong> not only generates the solutions to a quadratic equation, but can also tell us about the nature of the solutions. When we consider the <strong>discriminant<\/strong>, or the expression under the radical, [latex]{b}^{2}-4ac[\/latex], it tells us whether the solutions are real numbers or complex numbers and how many solutions of each type to expect.<\/p>\n<p>Let us explore how the discriminant affects the evaluation of [latex]\\sqrt{{{b}^{2}}-4ac}[\/latex] in the quadratic formula and how it helps to determine the solution set.<\/p>\n<ul>\n<li>If [latex]b^{2}-4ac>0[\/latex], then the number underneath the radical will be a positive value. You can always find the square root of a positive number, so evaluating the quadratic formula will result in two real solutions (one by adding the positive square root and one by subtracting it).<\/li>\n<li>If [latex]b^{2}-4ac=0[\/latex], then you will be taking the square root of\u00a0[latex]0[\/latex], which is\u00a0[latex]0[\/latex]. Since adding and subtracting\u00a0[latex]0[\/latex] both give the same result, the &#8220;[latex]\\pm[\/latex]&#8221; portion of the formula does not matter. There will be one real repeated solution.<\/li>\n<li>If [latex]b^{2}-4ac<0[\/latex], then the number underneath the radical will be a negative value. Since you cannot find the square root of a negative number using real numbers, there are no real solutions. However, you can use imaginary numbers. You will then have two complex solutions, one by adding the imaginary square root and one by subtracting it.<\/li>\n<\/ul>\n<p>The table below summarizes the relationship between the value of the discriminant and the solutions of a quadratic equation.<\/p>\n<table style=\"width: 60%;\" summary=\"A table with 5 rows and 2 columns. The entries in the first row are: Value of Discriminant and Results. The entries in the second row are: b squared minus four times a times c equals zero and One rational solution (double solution). The entries in the third row are: b squared minus four times a times c is greater than zero, perfect square and Two rational solutions. The entries in the fourth row are: b squared minus four times a times c is greater than zero, not a perfect square and Two irrational solutions. The entries in the fifth row are: b squared minus four times a times c is less than zero and Two complex solutions.\">\n<thead>\n<tr>\n<th>Value of Discriminant<\/th>\n<th>Results<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td>[latex]{b}^{2}-4ac=0[\/latex]<\/td>\n<td>One repeated rational solution<\/td>\n<\/tr>\n<tr>\n<td>[latex]{b}^{2}-4ac>0[\/latex], perfect square<\/td>\n<td>Two rational solutions<\/td>\n<\/tr>\n<tr>\n<td>[latex]{b}^{2}-4ac>0[\/latex], not a perfect square<\/td>\n<td>Two irrational solutions<\/td>\n<\/tr>\n<tr>\n<td>[latex]{b}^{2}-4ac<0[\/latex]<\/td>\n<td>Two complex solutions<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<div class=\"shaded textbox\">\n<h3>The Discriminant<\/h3>\n<p>For [latex]a{x}^{2}+bx+c=0[\/latex], where [latex]a[\/latex], [latex]b[\/latex], and [latex]c[\/latex] are real numbers, the <strong>discriminant<\/strong> is the expression under the radical in the quadratic formula: [latex]{b}^{2}-4ac[\/latex]. It tells us whether the solutions are real numbers or complex numbers and how many solutions of each type to expect.<\/p>\n<\/div>\n<div class=\"bcc-box bcc-info\">\n<h3>Example<\/h3>\n<p>Use the discriminant to determine how many and what kind of solutions the quadratic equation [latex]x^{2}-4x+10=0[\/latex]\u00a0has.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q116245\">Show Solution<\/span><\/p>\n<div id=\"q116245\" class=\"hidden-answer\" style=\"display: none\">\n<p>Evaluate [latex]b^{2}-4ac[\/latex]. First note that\u00a0[latex]a=1,b=\u22124[\/latex], and [latex]c=10[\/latex].<\/p>\n<p>[latex]\\begin{array}{l}b^{2}-4ac=\\left(-4\\right)^{2}-4\\left(1\\right)\\left(10\\right)=16-40=-24\\end{array}[\/latex]<\/p>\n<p>The result is a negative number. The discriminant is negative, so the quadratic equation has two complex solutions.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>Use the discriminant to find the nature of the solutions to the following quadratic equations:<\/p>\n<ol>\n<li>[latex]{x}^{2}+4x+4=0[\/latex]<\/li>\n<li>[latex]8{x}^{2}+14x+3=0[\/latex]<\/li>\n<li>[latex]3{x}^{2}-5x - 2=0[\/latex]<\/li>\n<li>[latex]3{x}^{2}-10x+15=0[\/latex]<\/li>\n<\/ol>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q497176\">Show Solution<\/span><\/p>\n<div id=\"q497176\" class=\"hidden-answer\" style=\"display: none\">\n<p>Calculate the discriminant [latex]{b}^{2}-4ac[\/latex] for each equation and state the expected type of solutions.<\/p>\n<ol>\n<li style=\"text-align: left;\">[latex]{x}^{2}+4x+4=0[\/latex] [latex]\\\\ {b}^{2}-4ac={\\left(4\\right)}^{2}-4\\left(1\\right)\\left(4\\right)=0[\/latex] [latex]\\text{There will be one repeated rational solution.}[\/latex]<\/li>\n<li style=\"text-align: left;\">[latex]8{x}^{2}+14x+3=0[\/latex][latex]\\\\ {b}^{2}-4ac={\\left(14\\right)}^{2}-4\\left(8\\right)\\left(3\\right)=100[\/latex] [latex]\\text{100 is a perfect square, so there will be two rational solutions.}[\/latex]<\/li>\n<li style=\"text-align: left;\">[latex]3{x}^{2}-5x - 2=0[\/latex][latex]\\\\ {b}^{2}-4ac={\\left(-5\\right)}^{2}-4\\left(3\\right)\\left(-2\\right)=49[\/latex] [latex]\\text{49 is a perfect square, so there will be two rational solutions.}[\/latex]<\/li>\n<li style=\"text-align: left;\">[latex]3{x}^{2}-10x+15=0[\/latex][latex]\\\\ {b}^{2}-4ac={\\left(-10\\right)}^{2}-4\\left(3\\right)\\left(15\\right)=-80[\/latex] [latex]\\text{There will be two complex solutions.}[\/latex]<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm2977\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=2977&theme=oea&iframe_resize_id=ohm2977&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<div class=\"textbox exercises\">\n<p>&nbsp;<\/p>\n<\/div>\n<p>We can summarize our results as follows:<\/p>\n<table>\n<tbody>\n<tr>\n<td>Discriminant<\/td>\n<td>Number and Type of Solutions<\/td>\n<td>Graph of\u00a0[latex]y=ax^2+bx+c[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]b^{2}-4ac<0[\/latex]<\/td>\n<td>two complex solutions<\/td>\n<td>will not cross the x-axis<\/td>\n<\/tr>\n<tr>\n<td>[latex]b^{2}-4ac=0[\/latex]<\/td>\n<td>one real repeated solution<\/td>\n<td>will touch x-axis once<\/td>\n<\/tr>\n<tr>\n<td>[latex]b^{2}-4ac>0[\/latex]<\/td>\n<td>\u00a0two real solutions<\/td>\n<td>\u00a0will cross x-axis twice<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>In the following video, we show more examples of how to use the discriminant to describe the type of solutions of a quadratic equation.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-1\" title=\"Ex:  The Discriminant\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/hSWs0VUyn1k?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<h2>Summary<\/h2>\n<p>The discriminant of the quadratic formula is the quantity under the radical, [latex]{{b}^{2}}-4ac[\/latex]. It determines the number and the type of solutions that a quadratic equation has. If the discriminant is positive, there are\u00a0[latex]2[\/latex] real solutions. If it is\u00a0[latex]0[\/latex], there is\u00a0[latex]1[\/latex] real repeated solution. If the discriminant is negative, there are\u00a0[latex]2[\/latex] complex solutions (but no real solutions).<\/p>\n<p>The discriminant can also tell us about the behavior of the graph of a quadratic equation in two variables.<\/p>\n<h2>Contribute!<\/h2>\n<div style=\"margin-bottom: 8px;\">Did you have an idea for improving this content? We\u2019d love your input.<\/div>\n<p><a style=\"font-size: 10pt; font-weight: 600; color: #077fab; text-decoration: none; border: 2px solid #077fab; border-radius: 7px; padding: 5px 25px; text-align: center; cursor: pointer; line-height: 1.5em;\" href=\"https:\/\/docs.google.com\/document\/d\/12KLcUeXJOwUh7viWyV3U6UXSmyJAEKrzYcQuzHXYuDU\" target=\"_blank\" rel=\"noopener\">Improve this page<\/a><a style=\"margin-left: 16px;\" href=\"https:\/\/docs.google.com\/document\/d\/1vy-T6DtTF-BbMfpVEI7VP_R7w2A4anzYZLXR8Pk4Fu4\" target=\"_blank\" rel=\"noopener\">Learn More<\/a><\/p>\n\n\t\t\t <section class=\"citations-section\" role=\"contentinfo\">\n\t\t\t <h3>Candela Citations<\/h3>\n\t\t\t\t\t <div>\n\t\t\t\t\t\t <div id=\"citation-list-15390\">\n\t\t\t\t\t\t\t <div class=\"licensing\"><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Original<\/div><ul class=\"citation-list\"><li>Revision and Adaptation. <strong>Provided by<\/strong>: Lumen Learning. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><\/ul><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Shared previously<\/div><ul class=\"citation-list\"><li>Ex: The Discriminant. <strong>Authored by<\/strong>: James Sousa (Mathispower4u.com) . <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/hSWs0VUyn1k\">https:\/\/youtu.be\/hSWs0VUyn1k<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><li>College Algebra. <strong>Authored by<\/strong>: Abramson, Jay et al.. <strong>Provided by<\/strong>: OpenStax. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"http:\/\/cnx.org\/contents\/9b08c294-057f-4201-9f48-5d6ad992740d@3.278:1\/Preface\">http:\/\/cnx.org\/contents\/9b08c294-057f-4201-9f48-5d6ad992740d@3.278:1\/Preface<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><\/ul><\/div>\n\t\t\t\t\t\t <\/div>\n\t\t\t\t\t <\/div>\n\t\t\t <\/section>","protected":false},"author":167848,"menu_order":14,"template":"","meta":{"_candela_citation":"[{\"type\":\"cc\",\"description\":\"Ex: The Discriminant\",\"author\":\"James Sousa (Mathispower4u.com) \",\"organization\":\"\",\"url\":\"https:\/\/youtu.be\/hSWs0VUyn1k\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"cc\",\"description\":\"College Algebra\",\"author\":\"Abramson, Jay et al.\",\"organization\":\"OpenStax\",\"url\":\" 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