Question

How do you solve `sqrt(x + 5) = sqrt(x - 3) + 2`?

Answer 1

Multiply by the conjugate to eliminate the squares
Use a linear algebra trick to eliminate the `sqrt(x-3)` term.
Square the remaining radical to solve.

Explanation:

Given: `sqrt(x + 5) = sqrt(x - 3) + 2`

Subtract `sqrt(x-3)` from both sides and mark it as equation [1]:

`sqrt(x + 5) - sqrt(x - 3) = 2" [1]"`

To eliminate the radicals on the left, multiply both sides by `sqrt(x + 5) + sqrt(x - 3)`:

`(x + 5) -(x - 3) = 2(sqrt(x + 5) + sqrt(x - 3))`

`x+ 5 -x + 3 = 2(sqrt(x + 5) + sqrt(x - 3))`

`8 = 2(sqrt(x + 5) + sqrt(x - 3))`

Divide both sides by 2 and mark as equation [2]:

`sqrt(x + 5) + sqrt(x - 3) = 4" [2]"`

Add equation [2] to equation [1]:

`2sqrt(x+5) = 6`

#sqrt(x+5) = 3

`x+5 = 9`

`x = 4`

Check:

`sqrt(4 + 5) = sqrt(4 - 3) + 2`

`sqrt9 = sqrt1+2`

`3 = 3`

This checks.