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

Apr 23, 2016

$x = 4$

#### Explanation:

We isolate the square root first so that we can simplify by squaring.

First subtract $1$ from both sides to get:

$\sqrt{x + 5} = x - 1$

Next square both sides (which may result in spurious solutions) to get:

$x + 5 = {x}^{2} - 2 x + 1$

Subtract $x + 5$ from both sides to get:

$0 = {x}^{2} - 3 x - 4 = \left(x - 4\right) \left(x + 1\right)$

Hence $x = 4$ or $x = - 1$

The solution $x = - 1$ of this quadratic is not a solution of the original equation:

$\sqrt{- 1 + 5} = \sqrt{4} = 2 \ne - 2 = - 1 - 1$

The other solution $x = 4$ is a solution of the original equation:

$\sqrt{4 + 5} = \sqrt{9} = 3 = 4 - 1$