Question

If \(\sqrt{x + 6\sqrt{2}} - \sqrt{x - 6\sqrt{2}} = 2\sqrt{2}\), then \(x\) equals:

• A. 2
• B. 17
• C. 11
• D. 13
• E. 12

Hint:

For equations of the form $\sqrt{a+\sqrt{b}}$, try to express the term under the square root as a perfect square of the form $(m+\sqrt{n})^2$.

Answer 1

The Correct Option is C

Step 1: Understanding the Concept:
Solving equations with square roots typically requires squaring both sides to eliminate the radicals.
Step 2: Key Formula or Approach:
Square the equation \( \sqrt{A} - \sqrt{B} = C \) using \( A + B - 2\sqrt{AB} = C^2 \).
Step 3: Detailed Explanation:
Given: \( \sqrt{x + 6\sqrt{2}} - \sqrt{x - 6\sqrt{2}} = 2\sqrt{2} \).
Squaring both sides:
\[ (x + 6\sqrt{2}) + (x - 6\sqrt{2}) - 2\sqrt{(x + 6\sqrt{2})(x - 6\sqrt{2})} = (2\sqrt{2})^2 \] \[ 2x - 2\sqrt{x^2 - (6\sqrt{2})^2} = 8 \] \[ 2x - 2\sqrt{x^2 - 72} = 8 \] \[ x - 4 = \sqrt{x^2 - 72} \] Squaring both sides again:
\[ (x - 4)^2 = x^2 - 72 \] \[ x^2 - 8x + 16 = x^2 - 72 \] \[ -8x = -88 \Rightarrow x = 11 \]
Step 4: Final Answer:
The value of x is 11.