Equation Involving Square Roots of Surds

Squaring both sides of $(x + 6\sqrt{2})^{\frac{1}{2}} - (x - 6\sqrt{2})^{\frac{1}{2}} = 2\sqrt{2}$: $(x + 6\sqrt{2}) + (x - 6\sqrt{2}) - 2\sqrt{(x + 6\sqrt{2})(x - 6\sqrt{2})} = (2\sqrt{2})^2$ $2x - 2\sqrt{x^2 - 72} = 8$ $x - 4 = \sqrt{x^2 - 72}$

Squaring both sides again: $(x - 4)^2 = x^2 - 72$ $x^2 - 8x + 16 = x^2 - 72$ $8x = 88 \implies x = 11$.