# What is the discriminant of x^2-9=0 and what does that mean?

Jul 30, 2015

In your case, $\Delta = 36$, which means that your equation has two distinct, rational solutions.

#### Explanation:

The general form of a quadratic equation is

$a {x}^{2} + b x + c = 0$

for which the discriminant is defined as

$\Delta = {b}^{2} - 4 \cdot a \cdot c$

In your case, $a = 1$, $b = 0$, and $c = - 9$, so the discriminant becomes

$\Delta = {0}^{2} - 4 \cdot 1 \cdot \left(- 9\right) = \textcolor{g r e e n}{36}$

A quadratic equation that has $\Delta > 0$ has two distinct, real solutions. Moreover, since $\Delta$ is a perfect square, two two solutions will be rational numbers.

The general form for the solutions of a quadratic equation is

color(blue)(x_(1,2) = (-b +- sqrt(Delta))/(2a)

In your case, those two solutions will be

${x}_{1 , 2} = \frac{0 \pm 6}{2} = \left\{\begin{matrix}{x}_{1} = 3 \\ {x}_{2} = - 3\end{matrix}\right.$

Note that these solutions could have easily been determined by

${x}^{2} - \textcolor{red}{\cancel{\textcolor{b l a c k}{9}}} + \textcolor{red}{\cancel{\textcolor{b l a c k}{9}}} = 9$

$\sqrt{{x}^{2}} = \sqrt{9} \implies {x}_{1 , 2} = \pm 3$