Question

\(\sqrt{x + 6\sqrt{2}} - \sqrt{x - 6\sqrt{2}} = 2\sqrt{2}.\)
Find \( x \).

Answer 1

Correct Answer: 11

Step 1: Let \(\sqrt{x + 6\sqrt{2}} = a\) and \(\sqrt{x - 6\sqrt{2}} = b\)

The equation becomes:
\(a - b = 2\sqrt{2}. \quad \text{(1)}\)

Square both sides:
\((a - b)^2 = (2\sqrt{2})^2.\)

Simplify:
\(a^2 - 2ab + b^2 = 8. \quad \text{(2)}\)

Step 2: Express \( a^2 \) and \( b^2 \) in terms of \( x \)

From the definitions:
\(a^2 = x + 6\sqrt{2}, \quad b^2 = x - 6\sqrt{2}.\)

Add \( a^2 + b^2 \):
\(a^2 + b^2 = (x + 6\sqrt{2}) + (x - 6\sqrt{2}) = 2x. \quad \text{(3)}\)

Subtract \( a^2 - b^2 \):
\(a^2 - b^2 = (x + 6\sqrt{2}) - (x - 6\sqrt{2}) = 12\sqrt{2}. \quad \text{(4)}\)

Step 3: Substitute into Equation (2)

From Equation (2):
\(a^2 + b^2 - 2ab = 8.\)

Substitute \( a^2 + b^2 = 2x \):
\(2x - 2ab = 8.\)

Solve for \( ab \):
\(ab = \frac{2x - 8}{2} = x - 4. \quad \text{(5)}\)

Step 4: Solve for \( a + b \) using \( (a - b)(a + b) = a^2 - b^2 \)

From Equation (1), \( a - b = 2\sqrt{2} \), and from Equation (4), \( a^2 - b^2 = 12\sqrt{2} \):
\((a - b)(a + b) = a^2 - b^2.\)

Substitute:
\((2\sqrt{2})(a + b) = 12\sqrt{2}.\)

Simplify:
\(a + b = 6. \quad \text{(6)}\)

Step 5: Solve for \( a \) and \( b \)

From the equations:
\(a - b = 2\sqrt{2}, \quad a + b = 6,\)

add the two equations:
\(2a = 6 + 2\sqrt{2}.\)

Solve for \( a \):
\(a = 3 + \sqrt{2}.\)

Subtract the two equations:
\(2b = 6 - 2\sqrt{2}.\)

Solve for \( b \):
\(b = 3 - \sqrt{2}.\)

Step 6: Use \( a^2 = x + 6\sqrt{2} \) to find \( x \)

Substitute \( a = 3 + \sqrt{2} \) into \( a^2 = x + 6\sqrt{2} \):
\((3 + \sqrt{2})^2 = x + 6\sqrt{2}.\)

Expand \( a^2 \):
\(a^2 = 9 + 6\sqrt{2} + 2 = 11 + 6\sqrt{2}.\)

Substitute:
\(11 + 6\sqrt{2} = x + 6\sqrt{2}.\)

Solve for \( x \):
\(x = 11.\)

Final Answer:
\(x = 11.\)