{"id":3268,"date":"2016-08-03T21:08:15","date_gmt":"2016-08-03T21:08:15","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/intermediatealgebra\/?post_type=chapter&#038;p=3268"},"modified":"2016-11-02T19:22:24","modified_gmt":"2016-11-02T19:22:24","slug":"read-quadratic-equations-with-complex-solutions","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/suny-suffolkccc-intermediatealgebra\/chapter\/read-quadratic-equations-with-complex-solutions\/","title":{"raw":"Classify Solutions to Quadratic Equations","rendered":"Classify Solutions to Quadratic Equations"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h3>Learning Objectives<\/h3>\r\n<ul>\r\n \t<li>Quadratic Equations with Complex Solutions\r\n<ul>\r\n \t<li>Use the quadratic formula to solve quadratic equations with complex solutions<\/li>\r\n \t<li>Connect complex solutions with the graph of a quadratic function that does not cross the x-axis.<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li>The Discriminant\r\n<ul>\r\n \t<li>Define the discriminant and use it to classify solutions to quadratic equations<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ul>\r\n<\/div>\r\nIn this lesson we will bring the concepts of solving quadratic equations and complex numbers together to find solutions to quadratic equations that are imaginary. \u00a0You will also learn about a really useful quality of the quadratic formula\u00a0that can help you know whether a quadratic equation has real or complex solutions.\r\n\r\nWe have seen two outcomes for solutions to\u00a0quadratic equations, either there was one or two real number solutions. We have also learned that it is possible to take the square root of a negative number by using imaginary numbers. Having this new knowledge allows us to explore one more possible outcome when we solve quadratic equations. Consider this equation:\r\n<p style=\"text-align: center;\">[latex]2x^2+3x+6=0[\/latex]<\/p>\r\n<p style=\"text-align: left;\">Using the Quadratic Formula to solve this equation, we first identify a, b, and c.<\/p>\r\n<p style=\"text-align: center;\">[latex]a = 2,b = 3,c = 6[\/latex]<\/p>\r\n<p style=\"text-align: left;\">We can place a, b and c into the quadratic formula and simplify to get the following result:<\/p>\r\n<p style=\"text-align: center;\">[latex]x=-\\frac{3}{4}+\\frac{\\sqrt{-39}}{4}, x=-\\frac{3}{4}-\\frac{\\sqrt{-39}}{4}[\/latex]<\/p>\r\n<p style=\"text-align: left;\">Up to this point, we would have said\u00a0that [latex]\\sqrt{-39}[\/latex] is not defined for real numbers and determine that this equation has no solutions. \u00a0But, now that we have defined the square root of a negative number, we can also define a solution to this equation as follows.<\/p>\r\n<p style=\"text-align: center;\">[latex]x=-\\frac{3}{4}+i\\frac{\\sqrt{39}}{4}, x=-\\frac{3}{4}-i\\frac{\\sqrt{39}}{4}[\/latex]<\/p>\r\n<p style=\"text-align: left;\">In this section we will practice simplifying and writing solutions to quadratic equations that are complex. \u00a0We will then present a technique for classifying whether the solution(s) to a quadratic equation will be complex, and how many solutions there will be.<\/p>\r\n<p style=\"text-align: left;\">In our first example we will work through the process of solving\u00a0a quadratic equation with\u00a0complex solutions. Take note that we be simplifying complex numbers - so if you need a review of how to rewrite the square root of a negative number as an imaginary number, now is a good time.<\/p>\r\n&nbsp;\r\n<div class=\"bcc-box bcc-info\">\r\n<h3>Example<\/h3>\r\nUse the Quadratic Formula to solve the equation [latex]x^{2}+2x=-5[\/latex].\r\n\r\n[reveal-answer q=\"654640\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"654640\"]First write the equation in standard form.\r\n\r\n[latex]\\begin{array}{r}x^{2}+2x=-5\\\\x^{2}+2x+5=0\\,\\,\\,\\,\\,\\\\\\\\a=1,b=2,c=5\\,\\,\\,\\end{array}[\/latex]\r\n\r\n[latex] \\begin{array}{r}{{x}^{2}}\\,\\,\\,+\\,\\,\\,2x\\,\\,\\,+\\,\\,\\,5\\,\\,\\,=\\,\\,\\,0\\\\\\downarrow\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\downarrow\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\downarrow\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\\\a{{x}^{2}}\\,\\,\\,+\\,\\,\\,bx\\,\\,\\,+\\,\\,\\,c\\,\\,\\,=\\,\\,\\,0\\end{array}[\/latex]\r\n\r\nSubstitute the values into the Quadratic Formula.\r\n\r\n[latex] x=\\frac{-b\\pm \\sqrt{{{b}^{2}}-4ac}}{2a}\\\\x=\\frac{-2\\pm \\sqrt{{{(2)}^{2}}-4(1)(5)}}{2(1)}[\/latex]\r\n\r\nSimplify, being careful to get the signs correct.\r\n\r\n[latex] x=\\frac{-2\\pm \\sqrt{4-20}}{2}[\/latex]\r\n\r\nSimplify some more.\r\n\r\n[latex] x=\\frac{-2\\pm \\sqrt{-16}}{2}[\/latex]\r\n\r\nSimplify the radical, but notice that the number under the radical symbol is negative! The square root of [latex]\u221216[\/latex] is imaginary. [latex] \\sqrt{-16}=4i[\/latex].\r\n\r\n[latex] x=\\frac{-2\\pm 4i}{2}[\/latex]\r\n\r\nSeparate and simplify to find the solutions to the quadratic equation.\r\n\r\n[latex]\\begin{array}{c}x=\\frac{-2+4i}{2}=\\frac{-1+2i}{1}\\cdot \\frac{2}{2}=-1+2i\\\\\\\\\\text{or}\\\\\\\\x=\\frac{-2-4i}{2}=\\frac{-1-2i}{1}\\cdot \\frac{2}{2}=-1-2i\\end{array}[\/latex]\r\n<h4>Answer<\/h4>\r\n[latex]x=-1+2i[\/latex] or [latex]-1-2i[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nWe can check these solutions in the original equation. Be careful when you expand the squares, and replace [latex]i^{2}[\/latex]\u00a0with [latex]-1[\/latex].\r\n<table>\r\n<tbody>\r\n<tr>\r\n<td>[latex]\\begin{array}{r}x=-1+2i\\\\x^{2}+2x=-5\\\\\\left(-1+2i\\right)^{2}+2\\left(-1+2i\\right)=-5\\\\1-4i+4i^{2}-2+4i=-5\\\\1-4i+4\\left(-1\\right)-2+4i=-5\\\\1-4-2=-5\\\\-5=-5\\end{array}[\/latex]<\/td>\r\n<td>[latex]\\begin{array}{r}x=-1-2i\\\\x^{2}+2x=-5\\\\\\left(-1-2i\\right)^{2}+2\\left(-1-2i\\right)=-5\\\\1+4i+4i^{2}-2-4i=-5\\\\1+4i+4\\left(-1\\right)-2-4i=-5\\\\1-4-2=-5\\\\-5=-5\\end{array}[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n&nbsp;\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\n<p id=\"fs-id1165135500790\">Use the quadratic formula to solve [latex]{x}^{2}+x+2=0[\/latex].<\/p>\r\n[reveal-answer q=\"144216\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"144216\"]First, we identify the coefficients: [latex]a=1,b=1[\/latex], and [latex]c=2[\/latex]. Substitute these values into the quadratic formula. [latex]\\begin{array}{l}x\\hfill&amp;=\\frac{-b\\pm \\sqrt{{b}^{2}-4ac}}{2a}\\hfill \\\\\\hfill&amp;=\\frac{-\\left(1\\right)\\pm \\sqrt{{\\left(1\\right)}^{2}-\\left(4\\right)\\cdot \\left(1\\right)\\cdot \\left(2\\right)}}{2\\cdot 1}\\hfill \\\\\\hfill&amp;=\\frac{-1\\pm \\sqrt{1 - 8}}{2}\\hfill \\\\ \\hfill&amp;=\\frac{-1\\pm \\sqrt{-7}}{2}\\hfill \\\\\\hfill&amp;=\\frac{-1\\pm i\\sqrt{7}}{2}\\hfill \\end{array}[\/latex]\r\n\r\nNow we can separate the expression [latex]\\frac{-1\\pm i\\sqrt{7}}{2}[\/latex] into two solutions:\r\n\r\n[latex]-\\frac{1}{2}+\\frac{ i\\sqrt{7}}{2}[\/latex]\r\n\r\n[latex]-\\frac{1}{2}-\\frac{ i\\sqrt{7}}{2}[\/latex]\r\n\r\n&nbsp;\r\n\r\nThe solutions to the equation are [latex]x=\\frac{-1+i\\sqrt{7}}{2}[\/latex] and [latex]x=\\frac{-1-i\\sqrt{7}}{2}[\/latex] or [latex]x=\\frac{-1}{2}+\\frac{i\\sqrt{7}}{2}[\/latex] and [latex]x=\\frac{-1}{2}-\\frac{i\\sqrt{7}}{2}[\/latex].[\/hidden-answer]\r\n\r\n<\/div>\r\nNow that we have had a little practice solving quadratic equations whose solutions are complex, we can explore a related\u00a0feature of quadratic functions. Consider the following function: [latex]f(x)=x^2+2x+3[\/latex]. \u00a0Recall that the x-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\n<p style=\"text-align: left;\">The function now looks like the type of quadratic equations we have been solving. \u00a0In the next example, we will solve this equation, then graph the original function and see that it has no x-intercepts.<\/p>\r\n\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nFind the x-intercepts of the quadratic function. [latex]f(x)=x^2+2x+3[\/latex]\r\n[reveal-answer q=\"698410\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"698410\"]\r\n\r\nThe x-intercepts of the function\u00a0[latex]f(x)=x^2+2x+3[\/latex] are found by setting it equal to zero, and solving for x since the y values of the x-intercepts are zero.\r\n\r\nFirst, identify a, b, c.\r\n\r\n[latex]\\begin{array}{ccc}x^2+2x+3=0\\\\a=1,b=2,c=3\\end{array}[\/latex]\r\n\r\nSubstitute these values into the quadratic formula.\r\n\r\n&nbsp;\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{c}x=\\frac{-b\\pm \\sqrt{{b}^{2}-4ac}}{2a}\\\\=\\frac{-2\\pm \\sqrt{{2}^{2}-4(1)(3)}}{2(1)}\\\\=\\frac{-2\\pm \\sqrt{4-12}}{2} \\\\=\\frac{-2\\pm \\sqrt{-8}}{2}\\\\\\frac{-2\\pm 2i\\sqrt{2}}{2} \\\\-1\\pm i\\sqrt{2}=-1+\\sqrt{2},-1-\\sqrt{2}\\end{array}[\/latex]<\/p>\r\nThe solutions to this equations are complex, therefore there are no x-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_3475\" align=\"aligncenter\" width=\"241\"]<img class=\"wp-image-3475\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/121\/2016\/08\/04190045\/Screen-Shot-2016-08-04-at-11.34.19-AM.png\" alt=\"Graph of quadratic function with the following points (-1,2), (-2,3), (0,3), (1,6), (-3,6). \" width=\"241\" height=\"249\" \/> Graph of quadratic function with no x-intercepts in the real numbers.[\/caption]\r\n\r\nNote how the graph does not cross the x-axis, therefore there are no real x-intercepts for this function.[\/hidden-answer]\r\n\r\n<\/div>\r\nThe following video gives another example of how to use the quadratic formula to find solutions to a quadratic equation that has complex solutions.\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. The table below\u00a0relates the value of the discriminant to 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=\"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 Answer[\/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]{x}^{2}+4x+4=0[\/latex][latex]{b}^{2}-4ac={\\left(4\\right)}^{2}-4\\left(1\\right)\\left(4\\right)=0[\/latex]. There will be one repeated rational solution.<\/li>\r\n \t<li>[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]. As [latex]100[\/latex] is a perfect square, there will be two rational solutions.<\/li>\r\n \t<li>[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]. As [latex]49[\/latex] is a perfect square, there will be two rational solutions.<\/li>\r\n \t<li>[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]. 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\nIn the Quadratic Formula, the expression underneath the radical symbol determines the number and type of solutions the formula will reveal. This expression, [latex]b^{2}-4ac[\/latex], is called the <strong>discriminant<\/strong> of the equation [latex]ax^{2}+bx+c=0[\/latex].\r\n\r\nLet\u2019s think about how 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 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 0, which is 0. Since adding and subtracting 0 both give the same result, the \"[latex]\\pm[\/late]\" 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\"]Evaluate [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}{c}b^{2}-4ac\\\\\\left(-4\\right)^{2}-4\\left(1\\right)\\left(10\\right)\\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[latex]16\u201340=\u221224[\/latex]\r\n<h4>Answer<\/h4>\r\nThe quadratic equation [latex]x^{2}-4x+10=0[\/latex]\u00a0has two complex solutions.\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nIn the last example, we will draw a correlation between the number and type of solutions to a quadratic equation and\u00a0the graph of it's corresponding function.\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nUse the following graphs of quadratic functions to determine how many and what type of solutions the corresponding quadratic equation [latex]f(x)=0[\/latex] will have. \u00a0Determine whether the discriminant will be greater than, less than, or equal to zero for each.\r\n\r\na.\r\n\r\n<img class=\"alignnone size-full wp-image-3479\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/121\/2016\/08\/04191147\/Screen-Shot-2016-08-04-at-12.10.26-PM.png\" alt=\"Screen Shot 2016-08-04 at 12.10.26 PM\" width=\"148\" height=\"135\" \/>\r\n\r\nb.\r\n\r\n<img class=\"alignnone size-full wp-image-3480\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/121\/2016\/08\/04191227\/Screen-Shot-2016-08-04-at-12.12.08-PM.png\" alt=\"Screen Shot 2016-08-04 at 12.12.08 PM\" width=\"154\" height=\"136\" \/>\r\n\r\nc.\r\n\r\n<img class=\"alignnone size-full wp-image-3481\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/121\/2016\/08\/04191433\/Screen-Shot-2016-08-04-at-12.14.18-PM.png\" alt=\"Screen Shot 2016-08-04 at 12.14.18 PM\" width=\"170\" height=\"140\" \/>\r\n[reveal-answer q=\"26060\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"26060\"]\r\n\r\na. This quadratic function does not touch or cross the x-axis, therefore the corresponding equation [latex]f(x)=0[\/latex] will have complex solutions. This implies that [latex]b^{2}-4ac&lt;0[\/latex].\r\n\r\nb. This quadratic function touches the x-axis exactly once, which implies there is one repeated solution to the equation [latex]f(x)=0[\/latex]. \u00a0We can then say that\u00a0[latex]b^{2}-4ac=0[\/latex]\r\n\r\nc. In our final graph, the quadratic function crosses the x-axis twice which tells us that there are two real number solutions to the equation [latex]f(x)=0[\/latex], and therefore\u00a0[latex]b^{2}-4ac&gt;0[\/latex].\r\n\r\n[\/hidden-answer]\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 Quadratic Function<\/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 to 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 2 real solutions. If it is 0, there is 1 real repeated solution. If the discriminant is negative, there are 2 complex solutions (but no real solutions).\r\n\r\nThe discriminant can also tell us about the behavior of the graph of a quadratic function.\r\n<h2>Summary<\/h2>\r\nQuadratic equations can have complex solutions. \u00a0Quadratic functions whose graphs\u00a0do not cross the x-axis will have complex solutions for [latex]f(x)=0[\/latex].\r\n\r\n&nbsp;","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Learning Objectives<\/h3>\n<ul>\n<li>Quadratic Equations with Complex Solutions\n<ul>\n<li>Use the quadratic formula to solve quadratic equations with complex solutions<\/li>\n<li>Connect complex solutions with the graph of a quadratic function that does not cross the x-axis.<\/li>\n<\/ul>\n<\/li>\n<li>The Discriminant\n<ul>\n<li>Define the discriminant and use it to classify solutions to quadratic equations<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<\/div>\n<p>In this lesson we will bring the concepts of solving quadratic equations and complex numbers together to find solutions to quadratic equations that are imaginary. \u00a0You will also learn about a really useful quality of the quadratic formula\u00a0that can help you know whether a quadratic equation has real or complex solutions.<\/p>\n<p>We have seen two outcomes for solutions to\u00a0quadratic equations, either there was one or two real number solutions. We have also learned that it is possible to take the square root of a negative number by using imaginary numbers. Having this new knowledge allows us to explore one more possible outcome when we solve quadratic equations. Consider this equation:<\/p>\n<p style=\"text-align: center;\">[latex]2x^2+3x+6=0[\/latex]<\/p>\n<p style=\"text-align: left;\">Using the Quadratic Formula to solve this equation, we first identify a, b, and c.<\/p>\n<p style=\"text-align: center;\">[latex]a = 2,b = 3,c = 6[\/latex]<\/p>\n<p style=\"text-align: left;\">We can place a, b and c into the quadratic formula and simplify to get the following result:<\/p>\n<p style=\"text-align: center;\">[latex]x=-\\frac{3}{4}+\\frac{\\sqrt{-39}}{4}, x=-\\frac{3}{4}-\\frac{\\sqrt{-39}}{4}[\/latex]<\/p>\n<p style=\"text-align: left;\">Up to this point, we would have said\u00a0that [latex]\\sqrt{-39}[\/latex] is not defined for real numbers and determine that this equation has no solutions. \u00a0But, now that we have defined the square root of a negative number, we can also define a solution to this equation as follows.<\/p>\n<p style=\"text-align: center;\">[latex]x=-\\frac{3}{4}+i\\frac{\\sqrt{39}}{4}, x=-\\frac{3}{4}-i\\frac{\\sqrt{39}}{4}[\/latex]<\/p>\n<p style=\"text-align: left;\">In this section we will practice simplifying and writing solutions to quadratic equations that are complex. \u00a0We will then present a technique for classifying whether the solution(s) to a quadratic equation will be complex, and how many solutions there will be.<\/p>\n<p style=\"text-align: left;\">In our first example we will work through the process of solving\u00a0a quadratic equation with\u00a0complex solutions. Take note that we be simplifying complex numbers &#8211; so if you need a review of how to rewrite the square root of a negative number as an imaginary number, now is a good time.<\/p>\n<p>&nbsp;<\/p>\n<div class=\"bcc-box bcc-info\">\n<h3>Example<\/h3>\n<p>Use the Quadratic Formula to solve the equation [latex]x^{2}+2x=-5[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q654640\">Show Solution<\/span><\/p>\n<div id=\"q654640\" class=\"hidden-answer\" style=\"display: none\">First write the equation in standard form.<\/p>\n<p>[latex]\\begin{array}{r}x^{2}+2x=-5\\\\x^{2}+2x+5=0\\,\\,\\,\\,\\,\\\\\\\\a=1,b=2,c=5\\,\\,\\,\\end{array}[\/latex]<\/p>\n<p>[latex]\\begin{array}{r}{{x}^{2}}\\,\\,\\,+\\,\\,\\,2x\\,\\,\\,+\\,\\,\\,5\\,\\,\\,=\\,\\,\\,0\\\\\\downarrow\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\downarrow\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\downarrow\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\\\a{{x}^{2}}\\,\\,\\,+\\,\\,\\,bx\\,\\,\\,+\\,\\,\\,c\\,\\,\\,=\\,\\,\\,0\\end{array}[\/latex]<\/p>\n<p>Substitute the values into the Quadratic Formula.<\/p>\n<p>[latex]x=\\frac{-b\\pm \\sqrt{{{b}^{2}}-4ac}}{2a}\\\\x=\\frac{-2\\pm \\sqrt{{{(2)}^{2}}-4(1)(5)}}{2(1)}[\/latex]<\/p>\n<p>Simplify, being careful to get the signs correct.<\/p>\n<p>[latex]x=\\frac{-2\\pm \\sqrt{4-20}}{2}[\/latex]<\/p>\n<p>Simplify some more.<\/p>\n<p>[latex]x=\\frac{-2\\pm \\sqrt{-16}}{2}[\/latex]<\/p>\n<p>Simplify the radical, but notice that the number under the radical symbol is negative! The square root of [latex]\u221216[\/latex] is imaginary. [latex]\\sqrt{-16}=4i[\/latex].<\/p>\n<p>[latex]x=\\frac{-2\\pm 4i}{2}[\/latex]<\/p>\n<p>Separate and simplify to find the solutions to the quadratic equation.<\/p>\n<p>[latex]\\begin{array}{c}x=\\frac{-2+4i}{2}=\\frac{-1+2i}{1}\\cdot \\frac{2}{2}=-1+2i\\\\\\\\\\text{or}\\\\\\\\x=\\frac{-2-4i}{2}=\\frac{-1-2i}{1}\\cdot \\frac{2}{2}=-1-2i\\end{array}[\/latex]<\/p>\n<h4>Answer<\/h4>\n<p>[latex]x=-1+2i[\/latex] or [latex]-1-2i[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>We can check these solutions in the original equation. Be careful when you expand the squares, and replace [latex]i^{2}[\/latex]\u00a0with [latex]-1[\/latex].<\/p>\n<table>\n<tbody>\n<tr>\n<td>[latex]\\begin{array}{r}x=-1+2i\\\\x^{2}+2x=-5\\\\\\left(-1+2i\\right)^{2}+2\\left(-1+2i\\right)=-5\\\\1-4i+4i^{2}-2+4i=-5\\\\1-4i+4\\left(-1\\right)-2+4i=-5\\\\1-4-2=-5\\\\-5=-5\\end{array}[\/latex]<\/td>\n<td>[latex]\\begin{array}{r}x=-1-2i\\\\x^{2}+2x=-5\\\\\\left(-1-2i\\right)^{2}+2\\left(-1-2i\\right)=-5\\\\1+4i+4i^{2}-2-4i=-5\\\\1+4i+4\\left(-1\\right)-2-4i=-5\\\\1-4-2=-5\\\\-5=-5\\end{array}[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>&nbsp;<\/p>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p id=\"fs-id1165135500790\">Use the quadratic formula to solve [latex]{x}^{2}+x+2=0[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q144216\">Show Answer<\/span><\/p>\n<div id=\"q144216\" class=\"hidden-answer\" style=\"display: none\">First, we identify the coefficients: [latex]a=1,b=1[\/latex], and [latex]c=2[\/latex]. Substitute these values into the quadratic formula. [latex]\\begin{array}{l}x\\hfill&=\\frac{-b\\pm \\sqrt{{b}^{2}-4ac}}{2a}\\hfill \\\\\\hfill&=\\frac{-\\left(1\\right)\\pm \\sqrt{{\\left(1\\right)}^{2}-\\left(4\\right)\\cdot \\left(1\\right)\\cdot \\left(2\\right)}}{2\\cdot 1}\\hfill \\\\\\hfill&=\\frac{-1\\pm \\sqrt{1 - 8}}{2}\\hfill \\\\ \\hfill&=\\frac{-1\\pm \\sqrt{-7}}{2}\\hfill \\\\\\hfill&=\\frac{-1\\pm i\\sqrt{7}}{2}\\hfill \\end{array}[\/latex]<\/p>\n<p>Now we can separate the expression [latex]\\frac{-1\\pm i\\sqrt{7}}{2}[\/latex] into two solutions:<\/p>\n<p>[latex]-\\frac{1}{2}+\\frac{ i\\sqrt{7}}{2}[\/latex]<\/p>\n<p>[latex]-\\frac{1}{2}-\\frac{ i\\sqrt{7}}{2}[\/latex]<\/p>\n<p>&nbsp;<\/p>\n<p>The solutions to the equation are [latex]x=\\frac{-1+i\\sqrt{7}}{2}[\/latex] and [latex]x=\\frac{-1-i\\sqrt{7}}{2}[\/latex] or [latex]x=\\frac{-1}{2}+\\frac{i\\sqrt{7}}{2}[\/latex] and [latex]x=\\frac{-1}{2}-\\frac{i\\sqrt{7}}{2}[\/latex].<\/p><\/div>\n<\/div>\n<\/div>\n<p>Now that we have had a little practice solving quadratic equations whose solutions are complex, we can explore a related\u00a0feature of quadratic functions. Consider the following function: [latex]f(x)=x^2+2x+3[\/latex]. \u00a0Recall that the x-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 style=\"text-align: left;\">The function now looks like the type of quadratic equations we have been solving. \u00a0In the next example, we will solve this equation, then graph the original function and see that it has no x-intercepts.<\/p>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>Find the x-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 Answer<\/span><\/p>\n<div id=\"q698410\" class=\"hidden-answer\" style=\"display: none\">\n<p>The x-intercepts of the function\u00a0[latex]f(x)=x^2+2x+3[\/latex] are found by setting it equal to zero, and solving for x since the y values of the x-intercepts are zero.<\/p>\n<p>First, identify a, b, c.<\/p>\n<p>[latex]\\begin{array}{ccc}x^2+2x+3=0\\\\a=1,b=2,c=3\\end{array}[\/latex]<\/p>\n<p>Substitute these values into the quadratic formula.<\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{c}x=\\frac{-b\\pm \\sqrt{{b}^{2}-4ac}}{2a}\\\\=\\frac{-2\\pm \\sqrt{{2}^{2}-4(1)(3)}}{2(1)}\\\\=\\frac{-2\\pm \\sqrt{4-12}}{2} \\\\=\\frac{-2\\pm \\sqrt{-8}}{2}\\\\\\frac{-2\\pm 2i\\sqrt{2}}{2} \\\\-1\\pm i\\sqrt{2}=-1+\\sqrt{2},-1-\\sqrt{2}\\end{array}[\/latex]<\/p>\n<p>The solutions to this equations are complex, therefore there are no x-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_3475\" style=\"width: 251px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" aria-describedby=\"caption-attachment-3475\" class=\"wp-image-3475\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/121\/2016\/08\/04190045\/Screen-Shot-2016-08-04-at-11.34.19-AM.png\" alt=\"Graph of quadratic function with the following points (-1,2), (-2,3), (0,3), (1,6), (-3,6).\" width=\"241\" height=\"249\" \/><\/p>\n<p id=\"caption-attachment-3475\" class=\"wp-caption-text\">Graph of quadratic function with no x-intercepts in the real numbers.<\/p>\n<\/div>\n<p>Note how the graph does not cross the x-axis, therefore there are no real x-intercepts for this function.<\/p><\/div>\n<\/div>\n<\/div>\n<p>The following video gives another example of how to use the quadratic formula to find solutions to a quadratic equation that has complex solutions.<\/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. The table below\u00a0relates the value of the discriminant to 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=\"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 Answer<\/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]{x}^{2}+4x+4=0[\/latex][latex]{b}^{2}-4ac={\\left(4\\right)}^{2}-4\\left(1\\right)\\left(4\\right)=0[\/latex]. There will be one repeated rational solution.<\/li>\n<li>[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]. As [latex]100[\/latex] is a perfect square, there will be two rational solutions.<\/li>\n<li>[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]. As [latex]49[\/latex] is a perfect square, there will be two rational solutions.<\/li>\n<li>[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]. 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>In the Quadratic Formula, the expression underneath the radical symbol determines the number and type of solutions the formula will reveal. This expression, [latex]b^{2}-4ac[\/latex], is called the <strong>discriminant<\/strong> of the equation [latex]ax^{2}+bx+c=0[\/latex].<\/p>\n<p>Let\u2019s think about how 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 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 0, which is 0. Since adding and subtracting 0 both give the same result, the &#8220;[latex]\\pm[\/late]\" portion of the formula doesn'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\">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}{c}b^{2}-4ac\\\\\\left(-4\\right)^{2}-4\\left(1\\right)\\left(10\\right)\\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<p>[latex]16\u201340=\u221224[\/latex]<\/p>\n<h4>Answer<\/h4>\n<p>The quadratic equation [latex]x^{2}-4x+10=0[\/latex]\u00a0has two complex solutions.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>In the last example, we will draw a correlation between the number and type of solutions to a quadratic equation and\u00a0the graph of it's corresponding function.<\/p>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>Use the following graphs of quadratic functions to determine how many and what type of solutions the corresponding quadratic equation [latex]f(x)=0[\/latex] will have. \u00a0Determine whether the discriminant will be greater than, less than, or equal to zero for each.<\/p>\n<p>a.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-3479\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/121\/2016\/08\/04191147\/Screen-Shot-2016-08-04-at-12.10.26-PM.png\" alt=\"Screen Shot 2016-08-04 at 12.10.26 PM\" width=\"148\" height=\"135\" \/><\/p>\n<p>b.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-3480\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/121\/2016\/08\/04191227\/Screen-Shot-2016-08-04-at-12.12.08-PM.png\" alt=\"Screen Shot 2016-08-04 at 12.12.08 PM\" width=\"154\" height=\"136\" \/><\/p>\n<p>c.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-3481\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/121\/2016\/08\/04191433\/Screen-Shot-2016-08-04-at-12.14.18-PM.png\" alt=\"Screen Shot 2016-08-04 at 12.14.18 PM\" width=\"170\" height=\"140\" \/><\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q26060\">Show Answer<\/span><\/p>\n<div id=\"q26060\" class=\"hidden-answer\" style=\"display: none\">\n<p>a. This quadratic function does not touch or cross the x-axis, therefore the corresponding equation [latex]f(x)=0[\/latex] will have complex solutions. This implies that [latex]b^{2}-4ac<0[\/latex].\n\nb. This quadratic function touches the x-axis exactly once, which implies there is one repeated solution to the equation [latex]f(x)=0[\/latex]. \u00a0We can then say that\u00a0[latex]b^{2}-4ac=0[\/latex]\n\nc. In our final graph, the quadratic function crosses the x-axis twice which tells us that there are two real number solutions to the equation [latex]f(x)=0[\/latex], and therefore\u00a0[latex]b^{2}-4ac>0[\/latex].<\/p>\n<\/div>\n<\/div>\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 Quadratic Function<\/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 to a quadratic equation.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-2\" 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 2 real solutions. If it is 0, there is 1 real repeated solution. If the discriminant is negative, there are 2 complex solutions (but no real solutions).<\/p>\n<p>The discriminant can also tell us about the behavior of the graph of a quadratic function.<\/p>\n<h2>Summary<\/h2>\n<p>Quadratic equations can have complex solutions. \u00a0Quadratic functions whose graphs\u00a0do not cross the x-axis will have complex solutions for [latex]f(x)=0[\/latex].<\/p>\n<p>&nbsp;<\/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-3268\">\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: Quadratic Formula - Complex Solutions. <strong>Authored by<\/strong>: James Sousa (Mathispower4u.com) . <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/11EwTcRMPn8\">https:\/\/youtu.be\/11EwTcRMPn8<\/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><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":21,"menu_order":4,"template":"","meta":{"_candela_citation":"[{\"type\":\"cc\",\"description\":\"Ex: Quadratic Formula - 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