Step 1: Define variables and square the given equation.
Let \(a = \sqrt{x + 6\sqrt{2}}\) and \(b = \sqrt{x - 6\sqrt{2}}\). The equation becomes:
\(a - b = 2\sqrt{2}. \quad \text{(1)}\)
Squaring both sides yields:
\((a - b)^2 = (2\sqrt{2})^2.
\(a^2 - 2ab + b^2 = 8. \quad \text{(2)}\)
Step 2: Express \( a^2 \) and \( b^2 \) in terms of \( x \).
From their definitions:
\(a^2 = x + 6\sqrt{2}\) and \(b^2 = x - 6\sqrt{2}\).
Adding them:
\(a^2 + b^2 = (x + 6\sqrt{2}) + (x - 6\sqrt{2}) = 2x. \quad \text{(3)}\)
Subtracting them:
\(a^2 - b^2 = (x + 6\sqrt{2}) - (x - 6\sqrt{2}) = 12\sqrt{2}. \quad \text{(4)}\)
Step 3: Substitute into Equation (2).
Equation (2) is \(a^2 + b^2 - 2ab = 8\). Substituting \(a^2 + b^2 = 2x\):
\(2x - 2ab = 8.\)
Solving for \( ab \):
\(ab = \frac{2x - 8}{2} = x - 4. \quad \text{(5)}\)
Step 4: Find \( a + b \) using the difference of squares.
We know \( (a - b)(a + b) = a^2 - b^2 \). Using Equation (1) \(a - b = 2\sqrt{2}\) and Equation (4) \(a^2 - b^2 = 12\sqrt{2}\):
\((2\sqrt{2})(a + b) = 12\sqrt{2}.\)
This simplifies to:
\(a + b = 6. \quad \text{(6)}\)
Step 5: Solve for \( a \) and \( b \).
We have a system of two linear equations:
\(a - b = 2\sqrt{2}\)
\(a + b = 6\)
Adding these equations gives \(2a = 6 + 2\sqrt{2}\), so:
\(a = 3 + \sqrt{2}.\)
Subtracting the first equation from the second gives \(2b = 6 - 2\sqrt{2}\), so:
\(b = 3 - \sqrt{2}.\)
Step 6: Find \( x \) using \( a^2 = x + 6\sqrt{2} \).
Substitute \( a = 3 + \sqrt{2} \) into \( a^2 = x + 6\sqrt{2} \):
\((3 + \sqrt{2})^2 = x + 6\sqrt{2}.\)
Expanding the left side:
\(9 + 6\sqrt{2} + 2 = 11 + 6\sqrt{2}.\)
So, the equation becomes:
\(11 + 6\sqrt{2} = x + 6\sqrt{2}.\)
Solving for \( x \):
\(x = 11.\)
Final Answer:
\(x = 11.\)