To solve the problem, we need to evaluate the expression \[ \frac{p^{2} - q^{2}}{2pq} \] given that \[ p = 1 + \sqrt{2} \quad \text{and} \quad q = 1 - \sqrt{2}. \] ### Step 1: Calculate \( p^2 \) and \( q^2 \) First, we will calculate \( p^2 \) and \( q^2 \). 1. **Calculating \( p^2 \)**: \[ p^2 = (1 + \sqrt{2})^2 = 1^2 + 2 \cdot 1 \cdot \sqrt{2} + (\sqrt{2})^2 = 1 + 2\sqrt{2} + 2 = 3 + 2\sqrt{2}. \] 2. **Calculating \( q^2 \)**: \[ q^2 = (1 - \sqrt{2})^2 = 1^2 - 2 \cdot 1 \cdot \sqrt{2} + (\sqrt{2})^2 = 1 - 2\sqrt{2} + 2 = 3 - 2\sqrt{2}. \] ### Step 2: Calculate \( p^2 - q^2 \) Now, we will find \( p^2 - q^2 \): \[ p^2 - q^2 = (3 + 2\sqrt{2}) - (3 - 2\sqrt{2}) = 3 + 2\sqrt{2} - 3 + 2\sqrt{2} = 4\sqrt{2}. \] ### Step 3: Calculate \( 2pq \) Next, we need to calculate \( 2pq \): \[ pq = (1 + \sqrt{2})(1 - \sqrt{2}) = 1^2 - (\sqrt{2})^2 = 1 - 2 = -1. \] Thus, \[ 2pq = 2 \cdot (-1) = -2. \] ### Step 4: Substitute into the expression Now we can substitute \( p^2 - q^2 \) and \( 2pq \) into the expression: \[ \frac{p^2 - q^2}{2pq} = \frac{4\sqrt{2}}{-2} = -2\sqrt{2}. \] ### Conclusion The final answer is: \[ \boxed{-2\sqrt{2}}. \] ### Explanation of Options Now, let's analyze the options provided: - **Option A: \( 2(2 + \sqrt{2}) \)**: This simplifies to \( 4 + 2\sqrt{2} \), which is positive and does not match our answer. - **Option B: \( -2(2 + \sqrt{2}) \)**: This simplifies to \( -4 - 2\sqrt{2} \), which is also not equal to our answer. - **Option C: \( 2\sqrt{2} \)**: This is positive and does not match our answer. - **Option D: \( -2\sqrt{2} \)**: This is exactly what we calculated. ### Revision Summary - To evaluate the expression \(\frac{p^2 - q^2}{2pq}\), calculate \(p^2\) and \(q^2\) separately. - Use the difference of squares to find \(p^2 - q^2\). - Calculate \(2pq\) using the product of \(p\) and \(q\). - Substitute the results into the expression to find the final answer. - The correct answer is \(-2\sqrt{2}\), corresponding to option D.