Question

the square root of (x-11) = the square root of (x) - 1

The following equation will require that you square both sides twice before all the radicals are eliminated.

Answer 1

sqrt(x - 11) = sqrt(x) - 1

Well you already have a hint. We square both sides:
x - 11 = x - 2sqrt(x) + 1

Simplify it first. And take all terms with radical sign to the right side of equation:
x - x - 11 - 1 = -2sqrt(x)
-12 = -2sqrt(x)
6 = sqrt(x)

Square both sides for the second time:
x = 36