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’s graph below: Show Does 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: [latex]x^2+2x+3=0[/latex] In the next example we will solve this equation. You 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. By setting the function equal to zero and using the quadratic formula to solve, you will see that the roots are complex numbers. ExampleFind the [latex]x[/latex]-intercepts of the quadratic function. [latex]f(x)=x^2+2x+3[/latex] Show Solution The [latex]x[/latex]-intercepts of the function [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. First, identify [latex]a[/latex], [latex]b[/latex], and [latex]c[/latex]. [latex]x^2+2x+3=0[/latex] [latex]a=1,b=2,c=3[/latex] Substitute these values into the quadratic formula. [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] The solutions to this equation are complex, therefore there are no [latex]x[/latex]-intercepts for the function [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: Graph of quadratic function with no [latex]x[/latex]-intercepts in the real numbers. Note how the graph does not cross the [latex]x[/latex]-axis, therefore there are no real [latex]x[/latex]-intercepts for this function. Try ItTry itUse an online graphing calculator to construct a quadratic function that has complex roots. The following video gives another example of how to use the quadratic formula to find complex solutions to a quadratic equation. The DiscriminantThe quadratic formula not only generates the solutions to a quadratic equation, it tells us about the nature of the solutions. When we consider the discriminant, 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 relates the value of the discriminant to the solutions of a quadratic equation. Value of DiscriminantResults[latex]{b}^{2}-4ac=0[/latex]One repeated rational solution[latex]{b}^{2}-4ac>0[/latex], perfect squareTwo rational solutions[latex]{b}^{2}-4ac>0[/latex], not a perfect squareTwo irrational solutions[latex]{b}^{2}-4ac<0[/latex]Two complex solutionsA General Note: The DiscriminantFor [latex]a{x}^{2}+bx+c=0[/latex], where [latex]a[/latex], [latex]b[/latex], and [latex]c[/latex] are real numbers, the discriminant 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. ExampleUse the discriminant to find the nature of the solutions to the following quadratic equations:
Show Solution Calculate the discriminant [latex]{b}^{2}-4ac[/latex] for each equation and state the expected type of solutions.
We have seen that a quadratic equation may have two real solutions, one real solution, or two complex solutions. Let’s summarize how the discriminant affects the evaluation of [latex] \sqrt{{{b}^{2}}-4ac}[/latex], and how it helps to determine the solution set.
ExampleUse the discriminant to determine how many and what kind of solutions the quadratic equation [latex]x^{2}-4x+10=0[/latex] has. Show Solution Evaluate [latex]b^{2}-4ac[/latex]. First note that [latex]a=1,b=−4[/latex], and [latex]c=10[/latex]. [latex]{c}b^{2}-4ac[/latex] [latex]\left(-4\right)^{2}-4\left(1\right)\left(10\right)[/latex] [latex]16–40=−24[/latex] The result is a negative number. The discriminant is negative, so [latex]x^{2}-4x+10=0[/latex] has two complex solutions. What are the 2 roots of a quadratic equation?For the quadratic equation ax2 + bx + c = 0, the expression b2 – 4ac is called the discriminant. The value of the discriminant shows how many roots f(x) has: - If b2 – 4ac > 0 then the quadratic function has two distinct real roots. - If b2 – 4ac = 0 then the quadratic function has one repeated real root.
What are the roots of a quadratic function?Roots of Quadratic Equation. The values of variables satisfying the given quadratic equation are called its roots. In other words, x = α is a root of the quadratic equation f(x), if f(α) = 0. The real roots of an equation f(x) = 0 are the x-coordinates of the points where the curve y = f(x) intersect the x-axis.
What is √ B² 4ac?HSA.REI.B.4b. The discriminant is the part of the quadratic formula underneath the square root symbol: b²-4ac. The discriminant tells us whether there are two solutions, one solution, or no solutions.
What are the roots of the quadratic equation 2x² √ 5x 2 0 Using the quadratic formula?Solution: Given, the quadratic equation is 2x² - √5x - 2 = 0. We have to determine the roots of the quadratic equation. Therefore, the roots are (√5 + √21)/4 and (√5 - √21)/4.
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