To solve the equation \[ \frac{\sqrt{2}}{x+\sqrt{2}} = \frac{1}{x-\sqrt{2}}, \] we will cross-multiply to eliminate the fractions. This means we will multiply the numerator of the left side by the denominator of the right side and set it equal to the numerator of the right side multiplied by the denominator of the left side. ### Step 1: Cross-Multiply Cross-multiplying gives us: \[ \sqrt{2} \cdot (x - \sqrt{2}) = 1 \cdot (x + \sqrt{2}). \] ### Step 2: Distribute Now, we will distribute on both sides: \[ \sqrt{2}x - 2 = x + \sqrt{2}. \] ### Step 3: Rearranging the Equation Next, we want to get all terms involving \(x\) on one side and constant terms on the other side. We can do this by subtracting \(x\) from both sides and adding \(2\) to both sides: \[ \sqrt{2}x - x = \sqrt{2} + 2. \] ### Step 4: Factor Out \(x\) Now, we can factor out \(x\) on the left side: \[ (\sqrt{2} - 1)x = \sqrt{2} + 2. \] ### Step 5: Solve for \(x\) To isolate \(x\), we divide both sides by \((\sqrt{2} - 1)\): \[ x = \frac{\sqrt{2} + 2}{\sqrt{2} - 1}. \] ### Step 6: Rationalizing the Denominator To simplify this expression, we can multiply the numerator and the denominator by the conjugate of the denominator, which is \((\sqrt{2} + 1)\): \[ x = \frac{(\sqrt{2} + 2)(\sqrt{2} + 1)}{(\sqrt{2} - 1)(\sqrt{2} + 1)}. \] Calculating the denominator first: \[ (\sqrt{2} - 1)(\sqrt{2} + 1) = 2 - 1 = 1. \] Now, calculating the numerator: \[ (\sqrt{2} + 2)(\sqrt{2} + 1) = \sqrt{2} \cdot \sqrt{2} + \sqrt{2} + 2\sqrt{2} + 2 = 2 + 3\sqrt{2} + 2 = 4 + 3\sqrt{2}. \] Thus, we have: \[ x = 4 + 3\sqrt{2}. \] ### Conclusion The value of \(x\) is \[ \boxed{4 + 3\sqrt{2}}. \] ### Explanation of Other Options - **Option B: \(3\sqrt{2} - 4\)**: This option is incorrect because it does not match our derived expression for \(x\). - **Option C: \(3 - 2\sqrt{2}\)**: This option is also incorrect as it does not correspond to the solution we found. - **Option D: \(4 + 2\sqrt{2}\)**: This option is incorrect as it does not match the derived expression for \(x\). ### Revision Summary - Cross-multiply to eliminate fractions in equations. - Distribute terms carefully and rearrange to isolate the variable. - Factor out the variable and solve for it. - Rationalize the denominator when necessary to simplify the expression. - Always check each option against the derived solution to confirm correctness.