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

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Answer 1 · 2016-04-22T16:00:34

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

$x = 8$

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

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

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

$0 = x^2-11x+24 = (x-3)(x-8)$

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:

$sqrt(8+1) = sqrt(9) = 3 = 8 - 5$

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