# How do you solve abs(x^2 - 9 )= x^2 - 9?

Jul 10, 2018

The solutions are $S = \left\{- 3 , 3\right\}$

#### Explanation:

The equation is

$| {x}^{2} - 9 | = {x}^{2} - 9$

Therefore,

$\left\{\begin{matrix}{x}^{2} - 9 = {x}^{2} - 9 \\ {x}^{2} - 9 = - {x}^{2} + 9\end{matrix}\right.$

$\iff$, $\left\{\begin{matrix}\emptyset \\ 2 {x}^{2} - 18 = 0\end{matrix}\right.$

$\iff$, $\left\{\begin{matrix}\emptyset \\ {x}^{2} - 9 = 0\end{matrix}\right.$

$\iff$, $\left\{\begin{matrix}\emptyset \\ \left(x + 3\right) \left(x - 3\right) = 0\end{matrix}\right.$

$\iff$, $\left\{\begin{matrix}x = - 3 \\ x = 3\end{matrix}\right.$

The solutions are $S = \left\{- 3 , 3\right\}$