# How do you solve sqrt(x + 7) = x - 5?

Sep 7, 2016

$x = 9$

#### Explanation:

Given:

$\sqrt{x + 7} = x - 5$

Square both sides of the equation to get:

$x + 7 = {\left(x - 5\right)}^{2} = {x}^{2} - 10 x + 25$

Subtracting $x + 7$ from both ends, we get:

$0 = {x}^{2} - 11 x - 18 = \left(x - 2\right) \left(x - 9\right)$

So the solutions of this derived quadratic equation are $x = 2$ and $x = 9$.

Any solutions of the original equation must be solutions of the derived quadratic equation, so $x = 2$ and $x = 9$ are the only possible solutions of the original equation.

However, note that squaring is not a one to one function, so solutions of the derived equation may not be solutions of the original one.

In fact, we find:

$\sqrt{\textcolor{red}{2} + 7} = \sqrt{9} = 3 \ne - 3 = \textcolor{red}{2} - 5$

So $x = 2$ is not a solution of the original equation.

On the other hand, we find:

$\sqrt{\textcolor{red}{9} + 7} = \sqrt{16} = 4 = \textcolor{red}{9} - 5$

So $x = 9$ is a solution of the original equation.