Recall that we find the [latex]y[/latex]-intercept of a quadratic by evaluating the function at an input of zero, and we find the [latex]x[/latex]-intercepts at locations where the output is zero. Notice that the number of [latex]x[/latex]-intercepts can vary depending upon the location of the graph.
Number of [latex]x[/latex]-intercepts of a parabola
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.
The quadratic formula for an equation in the form [latex]ax^2 + bx + c = 0[/latex] is:
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]. Substituting these values into the formula we have:
The 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].
Analysis of the Solution
This quadratic equation has only non-real solutions.
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.
Consider the following function: [latex]f(x)=x^2+2x+3[/latex], and it’s graph below:
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]
Find the [latex]x[/latex]-intercepts of the quadratic function.
[latex]f(x)=x^2+2x+3[/latex]
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.
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.
The Discriminant
The 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]. 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 relates the value of the discriminant to the solutions of a quadratic equation.
Value of Discriminant
Results
[latex]{b}^{2}-4ac=0[/latex]
One rational solution (double solution)
[latex]{b}^{2}-4ac>0[/latex], perfect square
Two rational solutions
[latex]{b}^{2}-4ac>0[/latex], not a perfect square
Two irrational solutions
[latex]{b}^{2}-4ac<0[/latex]
Two complex solutions
discriminant
For [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.
Use the discriminant to find the nature of the solutions to the following quadratic equations:
[latex]{x}^{2}+4x+4=0[/latex]
[latex]8{x}^{2}+14x+3=0[/latex]
[latex]3{x}^{2}-5x - 2=0[/latex]
[latex]3{x}^{2}-10x+15=0[/latex]
Calculate the discriminant [latex]{b}^{2}-4ac[/latex] for each equation and state the expected type of solutions.
[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.
[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.
[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.
[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.
