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# $x$-intercepts of a Quadratic Function

The $x$-intercepts of the function $f\left(x\right)=a{x}^{2}+bx+c,\text{\hspace{0.17em}}\text{\hspace{0.17em}}a\ne 0$ are the solutions of the quadratic equation $a{x}^{2}+bx+c=0$. The solutions of a quadratic equation of the form $a{x}^{2}+bx+c=0$ are given by the quadratic formula.

$x=\frac{-b\pm \sqrt{{b}^{2}-4ac}}{2a}$

Here, the expression ${b}^{2}-4ac$ is called the discriminant.

The discriminant can be used to confirm the number of $x$-intercepts and the type of solutions of the quadratic equation.

Value of Discriminant |
Type and Number of $x$
-intercepts (roots) |
Example of Graph of Related Function |

${b}^{2}-4ac>\mathrm{0;}$ ${b}^{2}-4ac$ is a perfect square |
$2$ real and rational | $f\left(x\right)={x}^{2}-x-6$ |

${b}^{2}-4ac>\mathrm{0;}$
${b}^{2}-4ac$ is not a perfect square |
$2$ real and irrational |
$f\left(x\right)={x}^{2}-3$ |

${b}^{2}-4ac=0$ | $1$ real |
$f\left(x\right)={(x-2)}^{2}$ |

${b}^{2}-4ac<0$ | No $x$-intercepts |
$f\left(x\right)={x}^{2}-4x+7$ |