Question

How do you solve `sqrt(x+4)-2=sqrt(x-12)`?

Answer 1

`x=21`

Explanation:

Given:

`sqrt(x+4)-2 = sqrt(x-12)`

Square both sides of the equation (noting that this may introduce extraneous solutions) to get:

`(x+4)-4sqrt(x+4)+4=(x-12)`

Subtract `x` from both sides and simplify to get;

`8-4sqrt(x+4) = -12`

Subtract `8` from both sides to get:

`-4sqrt(x+4) = -20`

DIvide both sides by `-4` to get:

`sqrt(x+4) = 5`

Square both sides to get:

`x+4 = 25`

Subtract `4` from both sides to get:

`x=21`

Finally we need to check that this is a solution of the original equation, since squaring both sides of an equation squashes the distinction between positive and negative quantities.

`sqrt(color(blue)(21)+4) - 2 = sqrt(25)-2 = 5-2 = 3 = sqrt(9) = sqrt(color(blue)(21)-12)`

So `x=21` is a solution of the original equation.