Question

If $(x + 6\sqrt{2})^{\frac{1}{2}} - (x - 6\sqrt{2})^{\frac{1}{2}} = 2\sqrt{2}$, then $x$ equals

Accepted Answer

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

Equation Involving Square Roots of Surds

Explanation

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