# How do you solve sqrt( x + 1) =x - 5?

Apr 22, 2016

Square both sides, solve the resulting quadratic, then check the solutions to find valid solution:

$x = 8$

#### Explanation:

First square both sides, noting that this may introduce spurious solutions, to get:

$x + 1 = {x}^{2} - 10 x + 25$

Subtract $x + 1$ from both sides to get:

$0 = {x}^{2} - 11 x + 24 = \left(x - 3\right) \left(x - 8\right)$

So $x = 3$ or $x = 8$

The solution $x = 3$ of this quadratic is spurious since $x - 5 = - 2 < 0$, so does not match the positive square root in the original equation.

The solution $x = 8$ is a valid solution of the original equation:

$\sqrt{8 + 1} = \sqrt{9} = 3 = 8 - 5$