{"id":198,"date":"2023-06-21T13:22:43","date_gmt":"2023-06-21T13:22:43","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/chapter\/complex-roots\/"},"modified":"2023-09-29T03:08:58","modified_gmt":"2023-09-29T03:08:58","slug":"complex-roots","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/chapter\/complex-roots\/","title":{"raw":"\u25aa   Complex Roots","rendered":"\u25aa   Complex Roots"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h3>Learning Outcomes<\/h3>\r\n<ul>\r\n \t<li>Find the complex roots\u00a0of a quadratic function using the quadratic formula<b>.<\/b><\/li>\r\n \t<li>Use the discriminant to determine whether a quadratic function has real or complex roots.<\/li>\r\n<\/ul>\r\n<\/div>\r\n<h2>Complex\u00a0Roots<\/h2>\r\nNow you will hopefully begin to understand why we introduced complex numbers at the beginning of this module. Consider the following function: [latex]f(x)=x^2+2x+3[\/latex], and it's graph below:\r\n<p style=\"text-align: center;\"><img class=\"alignnone size-medium wp-image-4477\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/1916\/2016\/11\/18034032\/desmos-graph-300x300.png\" alt=\"Graph of quadratic function with the following points (-1,2), (-2,3), (0,3), (1,6), (-3,6). \" width=\"300\" height=\"300\" \/><\/p>\r\nDoes this function have roots? It's probably obvious that this function does not cross the [latex]x[\/latex]-axis, therefore it doesn't have any [latex]x[\/latex]-intercepts. Recall that the [latex]x[\/latex]-intercepts of a function are found by setting the function equal to zero:\r\n<p style=\"text-align: center;\">[latex]x^2+2x+3=0[\/latex]<\/p>\r\nIn the next example we will solve this equation. \u00a0You will see that there are roots, but they are not [latex]x[\/latex]-intercepts because the function does not contain [latex](x,y)[\/latex] pairs that are on the [latex]x[\/latex]-axis. We call these complex roots.\r\n\r\nBy setting the function equal to zero and using the quadratic formula to solve, you will see that the roots are complex numbers.\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nFind the [latex]x[\/latex]-intercepts of the quadratic function. [latex]f(x)=x^2+2x+3[\/latex]\r\n[reveal-answer q=\"698410\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"698410\"]\r\n\r\nThe [latex]x[\/latex]-intercepts of the function\u00a0[latex]f(x)=x^2+2x+3[\/latex] are found by setting it equal to zero, and solving for [latex]x[\/latex] since the [latex]y[\/latex] values of the [latex]x[\/latex]-intercepts are zero.\r\n\r\nFirst, identify [latex]a[\/latex], [latex]b[\/latex], and [latex]c[\/latex].\r\n<p style=\"text-align: center;\">[latex]x^2+2x+3=0[\/latex]<\/p>\r\n<p style=\"text-align: center;\">[latex]a=1,b=2,c=3[\/latex]<\/p>\r\nSubstitute these values into the quadratic formula.\r\n<p style=\"text-align: center;\">[latex]\\begin{align}x&amp;=\\dfrac{-b\\pm \\sqrt{{b}^{2}-4ac}}{2a}\\\\[1mm]&amp;=\\dfrac{-2\\pm \\sqrt{{2}^{2}-4(1)(3)}}{2(1)}\\\\[1mm]&amp;=\\dfrac{-2\\pm \\sqrt{4-12}}{2} \\\\[1mm]&amp;=\\dfrac{-2\\pm \\sqrt{-8}}{2}\\\\[1mm]&amp;=\\dfrac{-2\\pm 2i\\sqrt{2}}{2} \\\\[1mm]&amp;=-1\\pm i\\sqrt{2}\\\\[1mm]x&amp;=-1+i\\sqrt{2},-1-i\\sqrt{2}\\end{align}[\/latex]<\/p>\r\nThe solutions to this equation are complex, therefore there are no [latex]x[\/latex]-intercepts for the function\u00a0[latex]f(x)=x^2+2x+3[\/latex] in the set of real numbers that can be plotted on the Cartesian Coordinate plane. The graph of the function is plotted on the Cartesian Coordinate plane below:\r\n\r\n[caption id=\"attachment_4477\" align=\"aligncenter\" width=\"300\"]<img class=\"wp-image-4477 size-medium\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/1916\/2016\/11\/18034032\/desmos-graph-300x300.png\" alt=\"Graph of quadratic function with the following points (-1,2), (-2,3), (0,3), (1,6), (-3,6). \" width=\"300\" height=\"300\" \/> Graph of quadratic function with no [latex]x[\/latex]-intercepts in the real numbers.[\/caption]Note how the graph does not cross the [latex]x[\/latex]-axis, therefore there are no real [latex]x[\/latex]-intercepts for this function.[\/hidden-answer]<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\n[ohm_question]121401[\/ohm_question]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try it<\/h3>\r\nUse an online graphing calculator to construct a quadratic function that has complex roots.\r\n\r\n<\/div>\r\nThe following video gives another example of how to use the quadratic formula to find complex solutions to a quadratic equation.\r\n\r\nhttps:\/\/youtu.be\/11EwTcRMPn8\r\n<h2>The Discriminant<\/h2>\r\nThe <strong>quadratic formula<\/strong> not only generates the solutions to a quadratic equation, it tells 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. In turn, we can then determine whether a quadratic function has real or complex roots. The table below\u00a0relates the value of the discriminant to the solutions of a quadratic equation.\r\n<table 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=\"textbox\">\r\n<h3>A General Note: 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=\"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>[latex]\\begin{align}{x}^{2}+4x+4=0&amp;&amp;{b}^{2}-4ac={\\left(4\\right)}^{2}-4\\left(1\\right)\\left(4\\right)=0\\end{align}\\hspace{4mm}[\/latex]There will be one repeated rational solution.<\/li>\r\n \t<li>[latex]\\begin{align}8{x}^{2}+14x+3=0&amp;&amp;{b}^{2}-4ac={\\left(14\\right)}^{2}-4\\left(8\\right)\\left(3\\right)=100\\end{align}\\hspace{4mm}[\/latex]As [latex]100[\/latex] is a perfect square, there will be two rational solutions.<\/li>\r\n \t<li>[latex]\\begin{align}3{x}^{2}-5x - 2=0&amp;&amp;{b}^{2}-4ac={\\left(-5\\right)}^{2}-4\\left(3\\right)\\left(-2\\right)=49\\end{align}\\hspace{4mm}[\/latex]As [latex]49[\/latex] is a perfect square, there will be two rational solutions.<\/li>\r\n \t<li>[latex]\\begin{align}3{x}^{2}-10x+15=0&amp;&amp;{b}^{2}-4ac={\\left(-10\\right)}^{2}-4\\left(3\\right)\\left(15\\right)=-80\\end{align}\\hspace{4mm}[\/latex]There will be two complex solutions.<\/li>\r\n<\/ol>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nWe have seen\u00a0that a quadratic equation may have two real solutions, one real solution, or two complex solutions.\r\n\r\nLet\u2019s summarize\u00a0how the discriminant affects the evaluation of [latex] \\sqrt{{{b}^{2}}-4ac}[\/latex], 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, so evaluating the Quadratic Formula will result in two real solutions (one by adding the 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 0, which is 0. Since adding and subtracting 0 both give the same result, the \"[latex]\\pm[\/latex]\" portion of the formula doesn't 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\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<p style=\"text-align: center;\">[latex]{c}b^{2}-4ac[\/latex]<\/p>\r\n<p style=\"text-align: center;\">[latex]\\left(-4\\right)^{2}-4\\left(1\\right)\\left(10\\right)[\/latex]<\/p>\r\n<p style=\"text-align: center;\">[latex]16\u201340=\u221224[\/latex]<\/p>\r\nThe result is a negative number. The discriminant is negative, so [latex]x^{2}-4x+10=0[\/latex] has two complex solutions.\r\n\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 height=\"300\"]35145[\/ohm_question]\r\n\r\n<\/div>","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Learning Outcomes<\/h3>\n<ul>\n<li>Find the complex roots\u00a0of a quadratic function using the quadratic formula<b>.<\/b><\/li>\n<li>Use the discriminant to determine whether a quadratic function has real or complex roots.<\/li>\n<\/ul>\n<\/div>\n<h2>Complex\u00a0Roots<\/h2>\n<p>Now you will hopefully begin to understand why we introduced complex numbers at the beginning of this module. Consider the following function: [latex]f(x)=x^2+2x+3[\/latex], and it&#8217;s graph below:<\/p>\n<p style=\"text-align: center;\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-medium wp-image-4477\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/1916\/2016\/11\/18034032\/desmos-graph-300x300.png\" alt=\"Graph of quadratic function with the following points (-1,2), (-2,3), (0,3), (1,6), (-3,6).\" width=\"300\" height=\"300\" \/><\/p>\n<p>Does this function have roots? It&#8217;s probably obvious that this function does not cross the [latex]x[\/latex]-axis, therefore it doesn&#8217;t have any [latex]x[\/latex]-intercepts. Recall that the [latex]x[\/latex]-intercepts of a function are found by setting the function equal to zero:<\/p>\n<p style=\"text-align: center;\">[latex]x^2+2x+3=0[\/latex]<\/p>\n<p>In the next example we will solve this equation. \u00a0You will see that there are roots, but they are not [latex]x[\/latex]-intercepts because the function does not contain [latex](x,y)[\/latex] pairs that are on the [latex]x[\/latex]-axis. We call these complex roots.<\/p>\n<p>By setting the function equal to zero and using the quadratic formula to solve, you will see that the roots are complex numbers.<\/p>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>Find the [latex]x[\/latex]-intercepts of the quadratic function. [latex]f(x)=x^2+2x+3[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q698410\">Show Solution<\/span><\/p>\n<div id=\"q698410\" class=\"hidden-answer\" style=\"display: none\">\n<p>The [latex]x[\/latex]-intercepts of the function\u00a0[latex]f(x)=x^2+2x+3[\/latex] are found by setting it equal to zero, and solving for [latex]x[\/latex] since the [latex]y[\/latex] values of the [latex]x[\/latex]-intercepts are zero.<\/p>\n<p>First, identify [latex]a[\/latex], [latex]b[\/latex], and [latex]c[\/latex].<\/p>\n<p style=\"text-align: center;\">[latex]x^2+2x+3=0[\/latex]<\/p>\n<p style=\"text-align: center;\">[latex]a=1,b=2,c=3[\/latex]<\/p>\n<p>Substitute these values into the quadratic formula.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{align}x&=\\dfrac{-b\\pm \\sqrt{{b}^{2}-4ac}}{2a}\\\\[1mm]&=\\dfrac{-2\\pm \\sqrt{{2}^{2}-4(1)(3)}}{2(1)}\\\\[1mm]&=\\dfrac{-2\\pm \\sqrt{4-12}}{2} \\\\[1mm]&=\\dfrac{-2\\pm \\sqrt{-8}}{2}\\\\[1mm]&=\\dfrac{-2\\pm 2i\\sqrt{2}}{2} \\\\[1mm]&=-1\\pm i\\sqrt{2}\\\\[1mm]x&=-1+i\\sqrt{2},-1-i\\sqrt{2}\\end{align}[\/latex]<\/p>\n<p>The solutions to this equation are complex, therefore there are no [latex]x[\/latex]-intercepts for the function\u00a0[latex]f(x)=x^2+2x+3[\/latex] in the set of real numbers that can be plotted on the Cartesian Coordinate plane. The graph of the function is plotted on the Cartesian Coordinate plane below:<\/p>\n<div id=\"attachment_4477\" style=\"width: 310px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" aria-describedby=\"caption-attachment-4477\" class=\"wp-image-4477 size-medium\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/1916\/2016\/11\/18034032\/desmos-graph-300x300.png\" alt=\"Graph of quadratic function with the following points (-1,2), (-2,3), (0,3), (1,6), (-3,6).\" width=\"300\" height=\"300\" \/><\/p>\n<p id=\"caption-attachment-4477\" class=\"wp-caption-text\">Graph of quadratic function with no [latex]x[\/latex]-intercepts in the real numbers.<\/p>\n<\/div>\n<p>Note how the graph does not cross the [latex]x[\/latex]-axis, therefore there are no real [latex]x[\/latex]-intercepts for this function.<\/p><\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm121401\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=121401&theme=oea&iframe_resize_id=ohm121401&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try it<\/h3>\n<p>Use an online graphing calculator to construct a quadratic function that has complex roots.<\/p>\n<\/div>\n<p>The following video gives another example of how to use the quadratic formula to find complex solutions to a quadratic equation.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-1\" title=\"Ex:  Quadratic Formula - Complex Solutions\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/11EwTcRMPn8?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<h2>The Discriminant<\/h2>\n<p>The <strong>quadratic formula<\/strong> not only generates the solutions to a quadratic equation, it tells 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. In turn, we can then determine whether a quadratic function has real or complex roots. The table below\u00a0relates the value of the discriminant to the solutions of a quadratic equation.<\/p>\n<table 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=\"textbox\">\n<h3>A General Note: 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=\"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>[latex]\\begin{align}{x}^{2}+4x+4=0&&{b}^{2}-4ac={\\left(4\\right)}^{2}-4\\left(1\\right)\\left(4\\right)=0\\end{align}\\hspace{4mm}[\/latex]There will be one repeated rational solution.<\/li>\n<li>[latex]\\begin{align}8{x}^{2}+14x+3=0&&{b}^{2}-4ac={\\left(14\\right)}^{2}-4\\left(8\\right)\\left(3\\right)=100\\end{align}\\hspace{4mm}[\/latex]As [latex]100[\/latex] is a perfect square, there will be two rational solutions.<\/li>\n<li>[latex]\\begin{align}3{x}^{2}-5x - 2=0&&{b}^{2}-4ac={\\left(-5\\right)}^{2}-4\\left(3\\right)\\left(-2\\right)=49\\end{align}\\hspace{4mm}[\/latex]As [latex]49[\/latex] is a perfect square, there will be two rational solutions.<\/li>\n<li>[latex]\\begin{align}3{x}^{2}-10x+15=0&&{b}^{2}-4ac={\\left(-10\\right)}^{2}-4\\left(3\\right)\\left(15\\right)=-80\\end{align}\\hspace{4mm}[\/latex]There will be two complex solutions.<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/div>\n<p>We have seen\u00a0that a quadratic equation may have two real solutions, one real solution, or two complex solutions.<\/p>\n<p>Let\u2019s summarize\u00a0how the discriminant affects the evaluation of [latex]\\sqrt{{{b}^{2}}-4ac}[\/latex], 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, so evaluating the Quadratic Formula will result in two real solutions (one by adding the 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 0, which is 0. Since adding and subtracting 0 both give the same result, the &#8220;[latex]\\pm[\/latex]&#8221; portion of the formula doesn&#8217;t 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<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 style=\"text-align: center;\">[latex]{c}b^{2}-4ac[\/latex]<\/p>\n<p style=\"text-align: center;\">[latex]\\left(-4\\right)^{2}-4\\left(1\\right)\\left(10\\right)[\/latex]<\/p>\n<p style=\"text-align: center;\">[latex]16\u201340=\u221224[\/latex]<\/p>\n<p>The result is a negative number. The discriminant is negative, so [latex]x^{2}-4x+10=0[\/latex] has two complex solutions.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm35145\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=35145&theme=oea&iframe_resize_id=ohm35145&show_question_numbers\" width=\"100%\" height=\"300\"><\/iframe><\/p>\n<\/div>\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-198\">\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>Question ID 121401. <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>. <strong>License Terms<\/strong>: IMathAS Community License CC-BY + GPL<\/li><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>Question ID 35145. <strong>Authored by<\/strong>: Smart,Jim. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em>. <strong>License Terms<\/strong>: IMathAS Community License CC-BY + GPL<\/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@5.2\">http:\/\/cnx.org\/contents\/9b08c294-057f-4201-9f48-5d6ad992740d@5.2<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em>. <strong>License Terms<\/strong>: Download for free at http:\/\/cnx.org\/contents\/9b08c294-057f-4201-9f48-5d6ad992740d@5.2<\/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":395986,"menu_order":35,"template":"","meta":{"_candela_citation":"[{\"type\":\"original\",\"description\":\"Question ID 121401\",\"author\":\"\",\"organization\":\"Lumen Learning\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"IMathAS Community License CC-BY + GPL\"},{\"type\":\"cc\",\"description\":\"Question ID 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