Question

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

Answer 1

`x=9`

Explanation:

Given:

`sqrt(x+7) = x-5`

Square both sides of the equation to get:

`x+7 = (x-5)^2 = x^2-10x+25`

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

`0 = x^2-11x-18 = (x-2)(x-9)`

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(color(red)(2)+7) = sqrt(9) = 3 != -3 = color(red)(2)-5`

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

On the other hand, we find:

`sqrt(color(red)(9)+7) = sqrt(16) = 4 = color(red)(9)-5`

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