To solve the equation \(\frac{\sqrt{2}}{x+\sqrt{2}}=\frac{1}{x-\sqrt{2}}\), we will follow a step-by-step approach to isolate \(x\) and find its value. ### Step 1: Cross-Multiply We start by cross-multiplying the fractions to eliminate the denominators. This gives us: \[ \sqrt{2} \cdot (x - \sqrt{2}) = 1 \cdot (x + \sqrt{2}) \] ### Step 2: Distribute Next, we distribute both sides: \[ \sqrt{2}x - 2 = x + \sqrt{2} \] ### Step 3: Rearrange the Equation Now, 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: Rationalize the Denominator To simplify this expression, we can rationalize the denominator. We do this by multiplying 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: \[ (\sqrt{2} - 1)(\sqrt{2} + 1) = 2 - 1 = 1 \] Now, calculating the numerator: \[ (\sqrt{2} + 2)(\sqrt{2} + 1) = \sqrt{2} \cdot \sqrt{2} + \sqrt{2} \cdot 1 + 2 \cdot \sqrt{2} + 2 \cdot 1 = 2 + \sqrt{2} + 2\sqrt{2} + 2 = 4 + 3\sqrt{2} \] Thus, we have: \[ x = 4 + 3\sqrt{2} \] ### Step 7: Compare with Options Now, we compare our result \(x = 4 + 3\sqrt{2}\) with the provided options: - A. \(3\sqrt{2} + 4\) (This is equivalent to \(4 + 3\sqrt{2}\)) - B. \(3\sqrt{2} - 4\) - C. \(3 - 2\sqrt{2}\) - D. \(4 + 2\sqrt{2}\) The correct option is **A** (which is equivalent to our result). ### Explanation of Incorrect Options - **B. \(3\sqrt{2} - 4\)**: This option is incorrect because it subtracts \(4\) instead of adding it, which does not match our derived expression. - **C. \(3 - 2\sqrt{2}\)**: This option is incorrect as it does not contain the correct terms or coefficients derived from our calculations. - **D. \(4 + 2\sqrt{2}\)**: This option is incorrect because it has the wrong coefficient for \(\sqrt{2}\); our result has \(3\sqrt{2}\), not \(2\sqrt{2}\). ### Revision Summary - To solve the equation, cross-multiply to eliminate fractions. - Rearrange the equation to isolate \(x\). - Rationalize the denominator to simplify the expression. - Compare the final result with the provided options to find the correct answer. The correct answer is **A. \(3\sqrt{2} + 4\)**.