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

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Answer 1 · 2016-04-23T22:51:46

$x=4$

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:

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

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

$0 = x^2-3x-4 = (x-4)(x+1)$

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

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

$sqrt(-1+5) = sqrt(4) = 2 != -2 = -1-1$

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

$sqrt(4+5) = sqrt(9) = 3 = 4 - 1$

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