Question

How do you solve `sqrt(x+5 ) + sqrt(x-5) = 10`?

Answer 1

Rearrange and square a couple of times to find:

`x = 101/4`

Explanation:

Subtract `sqrt(x-5)` from both sides to get:

`sqrt(x+5) = 10 - sqrt(x-5)`

Square both sides to get:

`x+5 = 100 - 20sqrt(x-5) + (x-5)`

Add `20sqrt(x-5)` to both sides to get:

`20sqrt(x-5) + x + 5 = 100 + x - 5`

Subtract `x + 5` from both sides to get:

`20sqrt(x-5) = 100 - 10 = 90`

Divide both sides by `20` to get:

`sqrt(x-5) = 90/20 = 9/2`

Square both sides to get:

`x - 5 = (9/2)^2 = 81/4`

Add `5` to both sides to get:

`x = 81/4 + 5 = (81+20)/4 = 101/4`

Since we have squared the equation a couple of times, we should check that the solution is correct and not spurious:

Let `x = 101/4`

Then:

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

`=sqrt(101/4+5)+sqrt(101/4-5)`

`=sqrt(121/4) + sqrt(81/4)`

`=11/2 + 9/2 = 20/2 = 10`