Question

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

Answer 1

`x=5`

Explanation:

Given:

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

Squaring both sides, we get:

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

That is:

`2x-2sqrt(x^2-16) = 4`

Divide both sides by `2` to get:

`x-sqrt(x^2-16) = 2`

Add `sqrt(x^2-16)-2` to both sides to get:

`x-2 = sqrt(x^2-16)`

Square both sides to get:

`x^2-4x+4 = x^2-16`

Add `-x^2+4x+16` to both sides to get:

`20 = 4x`

Transpose and divide both sides by `4` to get:

`x = 5`

Since we have squared both sides of the equation - which not a reversible operation - we need to check that this solution we have reached is a solution of the original equation.

We find:

`sqrt((color(blue)(5))+4)-sqrt((color(blue)(5))-4) = sqrt(9)-sqrt(1) = 3-1 = 2`

So `x=5` is a valid solution.