{"id":4154,"date":"2024-09-18T15:30:40","date_gmt":"2024-09-18T15:30:40","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/?post_type=chapter&p=4154"},"modified":"2025-08-14T01:09:24","modified_gmt":"2025-08-14T01:09:24","slug":"complex-numbers-and-operations-learn-it-5","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/chapter\/complex-numbers-and-operations-learn-it-5\/","title":{"raw":"Complex Numbers and Operations: Learn It 5","rendered":"Complex Numbers and Operations: Learn It 5"},"content":{"raw":"

Complex\u00a0Roots<\/h2>\r\nRecall that we find the [latex]y[\/latex]<\/span>-intercept of a quadratic by evaluating the function at an input of zero, and we find the [latex]x[\/latex]<\/span>-intercepts at locations where the output is zero. Notice\u00a0that the number of [latex]x[\/latex]<\/span>-intercepts can vary depending upon the location of the graph.\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"975\"]\"Three Number of [latex]x[\/latex]<\/span>-intercepts of a parabola[\/caption]While factoring is often the first method we try when solving for [latex]x[\/latex]-intercepts, it's not always possible or practical. Some quadratic equations cannot be easily factored, especially those with irrational or complex roots. In these cases, you'll need a more powerful tool: the quadratic formula.\r\n\r\n
\r\n

The quadratic formula for an equation in the form [latex]ax^2 + bx + c = 0[\/latex] is:<\/p>\r\n

[latex]x = \\frac{-b \\pm \\sqrt{b^2 - 4ac}}{2a}[\/latex]<\/p>\r\n

Where:<\/p>\r\n\r\n

    \r\n \t
  • [latex]a[\/latex] is the coefficient of [latex]x^2[\/latex]<\/li>\r\n \t
  • [latex]b[\/latex] is the coefficient of [latex]x[\/latex]<\/li>\r\n \t
  • [latex]c[\/latex] is the constant term<\/li>\r\n \t
  • [latex]\\pm[\/latex] means we consider both addition and subtraction<\/li>\r\n<\/ul>\r\n<\/section>
    Solve [latex]{x}^{2}+x+2=0[\/latex] using the quadratic formula.[reveal-answer q=\"757696\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"757696\"]Let\u2019s begin by writing the quadratic formula:
    [latex]x=\\dfrac{-b\\pm \\sqrt{{b}^{2}-4ac}}{2a}[\/latex]<\/center>When applying the quadratic formula, we identify the coefficients [latex]a[\/latex], [latex]b[\/latex], and [latex]c[\/latex]. For the equation [latex]{x}^{2}+x+2=0[\/latex], we have [latex]a=1[\/latex], [latex]b=1[\/latex], and [latex]c=2[\/latex].\u00a0Substituting these values into the formula we have:\r\n

    [latex]\\begin{align}x&=\\dfrac{-b\\pm \\sqrt{{b}^{2}-4ac}}{2a} \\\\[1.5mm] &=\\dfrac{-1\\pm \\sqrt{{1}^{2}-4\\cdot 1\\cdot \\left(2\\right)}}{2\\cdot 1} \\\\[1.5mm] &=\\dfrac{-1\\pm \\sqrt{1 - 8}}{2} \\\\[1.5mm] &=\\dfrac{-1\\pm \\sqrt{-7}}{2} \\\\[1.5mm] &=\\dfrac{-1\\pm i\\sqrt{7}}{2} \\end{align}[\/latex]<\/p>\r\nThe solutions to the equation are [latex]x=\\dfrac{-1+i\\sqrt{7}}{2}[\/latex] and [latex]x=\\dfrac{-1-i\\sqrt{7}}{2}[\/latex] or [latex]x=-\\dfrac{1}{2}+\\dfrac{i\\sqrt{7}}{2}[\/latex] and [latex]x=-\\dfrac{1}{2}-\\dfrac{i\\sqrt{7}}{2}[\/latex].\r\n\r\nAnalysis of the Solution<\/strong>\r\n\r\nThis quadratic equation has only non-real solutions.\r\n\r\n[\/hidden-answer]\r\n\r\n<\/section>Now you will hopefully begin to understand why we introduced complex numbers at the beginning of this module. The example we just solved demonstrates a crucial concept in algebra: not all quadratic equations have real solutions. When the discriminant ([latex]b^2 - 4ac[\/latex]) is negative, we encounter complex roots. These complex roots always come in conjugate pairs, meaning if [latex]a + bi[\/latex] is a solution, then [latex]a - bi[\/latex] is also a solution.\r\n\r\n

    Consider the following function: [latex]f(x)=x^2+2x+3[\/latex], and it's graph below:\r\n\r\n[caption id=\"attachment_4477\" align=\"alignnone\" width=\"300\"]\"Graph Graph of a quadratic function[\/caption]\r\n\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

    [latex]x^2+2x+3=0[\/latex]<\/p>\r\n\r\n<\/section>

    Find the [latex]x[\/latex]-intercepts of the quadratic function.\r\n

    [latex]f(x)=x^2+2x+3[\/latex]<\/p>\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

    [latex]x^2+2x+3=0[\/latex]<\/p>\r\n

    [latex]a=1,b=2,c=3[\/latex]<\/p>\r\nSubstitute these values into the quadratic formula.\r\n

    [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>\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\"]\"Graph 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]<\/section>

    \r\n
    \r\n

    The Discriminant<\/h3>\r\nThe 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 discriminant<\/strong>, or the expression under the radical, [latex]{b}^{2}-4ac[\/latex]. The discriminant tells us whether the solutions are real numbers or complex numbers as well as 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\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n
    Value of Discriminant<\/th>\r\nResults<\/th>\r\n<\/tr>\r\n<\/thead>\r\n
    [latex]{b}^{2}-4ac=0[\/latex]<\/td>\r\nOne rational solution (double solution)<\/td>\r\n<\/tr>\r\n
    [latex]{b}^{2}-4ac>0[\/latex], perfect square<\/td>\r\nTwo rational solutions<\/td>\r\n<\/tr>\r\n
    [latex]{b}^{2}-4ac>0[\/latex], not a perfect square<\/td>\r\nTwo irrational solutions<\/td>\r\n<\/tr>\r\n
    [latex]{b}^{2}-4ac<0[\/latex]<\/td>\r\nTwo complex solutions<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/section><\/div>\r\n
    \r\n

    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 discriminant<\/strong> is the expression under the radical in the quadratic formula:\r\n\r\n \r\n

    [latex]{b}^{2}-4ac[\/latex].<\/p>\r\n \r\n\r\nIt tells us whether the solutions are real numbers or complex numbers and how many solutions of each type to expect.\r\n\r\n<\/section>

    Use the discriminant to find the nature of the solutions to the following quadratic equations:\r\n
      \r\n \t
    1. [latex]{x}^{2}+4x+4=0[\/latex]<\/li>\r\n \t
    2. [latex]8{x}^{2}+14x+3=0[\/latex]<\/li>\r\n \t
    3. [latex]3{x}^{2}-5x - 2=0[\/latex]<\/li>\r\n \t
    4. [latex]3{x}^{2}-10x+15=0[\/latex]<\/li>\r\n<\/ol>\r\n[reveal-answer q=\"229118\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"229118\"]\r\n\r\nCalculate the discriminant [latex]{b}^{2}-4ac[\/latex] for each equation and state the expected type of solutions.\r\n
        \r\n \t
      1. [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 rational double solution.<\/li>\r\n \t
      2. [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
      3. [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
      4. [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<\/section>
        [ohm2_question hide_question_numbers=1]18968[\/ohm2_question]<\/section>
        [ohm2_question hide_question_numbers=1]18970[\/ohm2_question]<\/section><\/section>","rendered":"

        Complex\u00a0Roots<\/h2>\n

        Recall that we find the [latex]y[\/latex]<\/span>-intercept of a quadratic by evaluating the function at an input of zero, and we find the [latex]x[\/latex]<\/span>-intercepts at locations where the output is zero. Notice\u00a0that the number of [latex]x[\/latex]<\/span>-intercepts can vary depending upon the location of the graph.<\/p>\n

        \"Three
        Number of [latex]x[\/latex]<\/span>-intercepts of a parabola<\/figcaption><\/figure>\n

        While factoring is often the first method we try when solving for [latex]x[\/latex]-intercepts, it’s not always possible or practical. Some quadratic equations cannot be easily factored, especially those with irrational or complex roots. In these cases, you’ll need a more powerful tool: the quadratic formula.<\/p>\n

        \n

        The quadratic formula for an equation in the form [latex]ax^2 + bx + c = 0[\/latex] is:<\/p>\n

        [latex]x = \\frac{-b \\pm \\sqrt{b^2 - 4ac}}{2a}[\/latex]<\/p>\n

        Where:<\/p>\n

          \n
        • [latex]a[\/latex] is the coefficient of [latex]x^2[\/latex]<\/li>\n
        • [latex]b[\/latex] is the coefficient of [latex]x[\/latex]<\/li>\n
        • [latex]c[\/latex] is the constant term<\/li>\n
        • [latex]\\pm[\/latex] means we consider both addition and subtraction<\/li>\n<\/ul>\n<\/section>\n
          Solve [latex]{x}^{2}+x+2=0[\/latex] using the quadratic formula.<\/p>\n