Square both sides of the equation. This cancels out the outermost radical sign on each side. On the next line, rewrite the equation leaving out those outermost radicals.
Example:
sqrt(x + sqrt(x - 3)) = sqrt(2x - 3)
(sqrt(x + sqrt(x - 3)))^2 = (sqrt(2x - 3))^2
x + sqrt(x - 3) = 2x - 3
Isolate the remaining radical on one side of the equation by adding or subtracting all other terms until they cancel out. Be sure to do the same thing to both sides of the equation.
Example:
x + sqrt(x - 3) = 2x - 3
-x -x
sqrt(x - 3) = x - 3
Square both sides of the equation again. Apply the FOIL method (First, Outside, Inside, Last) or the distributive property as needed for multiplying an expression by itself.
Example:
sqrt(x - 3) = x - 3
(sqrt(x - 3))^2 = (x - 3)^2
On the right-hand side, all the ^2 does is get rid of the sqrt, but on the left-hand side, you must use FOIL and combine like terms.
x - 3 = (x - 3)(x - 3)
x - 3 = x^2 - 3x - 3x + 9
x - 3 = x^2 - 6x + 9
Add or subtract terms from the shorter side until it is equal to zero.
Example:
x - 3 = x^2 - 6x + 9
-x -x
-3 = x^2 - 7x + 9
+3 +3
0 = x^2 - 7x + 12
Solve the quadratic equation using your favorite method.
Example:
Using factoring and setting both expressions equal to 0.
0 = x^2 - 7x + 12
0 = (x - 3)(x - 4)
x - 3 = 0
x = 3
x - 4 = 0
x = 4
Check all solutions by plugging them into the original equation one at a time. Sometimes, due to squaring multiple times, you may end up with extra answers, so this step allows you to find out which answers are valid.
Example:
x = 3 and x = 4
sqrt(3 + sqrt(3 - 3)) = sqrt(2*3 - 3)
sqrt(3 + sqrt(0)) = sqrt(6 - 3)
sqrt(3 + 0) = sqrt(3)
sqrt(3) = sqrt(3)
This answer works.
sqrt(4 + sqrt(4 - 3)) = sqrt(2*4 - 3)
sqrt(4 + sqrt(1)) = sqrt(8 - 3)
sqrt(4 + 1) = sqrt(5)
sqrt(5) = sqrt(5)
This answer also works. x = 3 and x = 4 are the answers.