Given that \(p = 1 + \sqrt{2}\) and \(q = 1 - \sqrt{2}\), evaluate \(\frac{p^{2} - q^{2}}{2pq}\).
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Correct Answer: Option D
Explanation:
\(\frac{p^{2} - q^{2}}{2pq} = \frac{(p + q)(p - q)}{2pq}\)
= \(\frac{(1 + \sqrt{2} - (1 - \sqrt{2}))(1 + \sqrt{2} + 1 - \sqrt{2})}{2(1 + \sqrt{2})(1 - \sqrt{2})}\)
= \(\frac{(2\sqrt{2})(2)}{-2}\)
= \(-2\sqrt{2}\)
\(\frac{p^{2} - q^{2}}{2pq} = \frac{(p + q)(p - q)}{2pq}\)
= \(\frac{(1 + \sqrt{2} - (1 - \sqrt{2}))(1 + \sqrt{2} + 1 - \sqrt{2})}{2(1 + \sqrt{2})(1 - \sqrt{2})}\)
= \(\frac{(2\sqrt{2})(2)}{-2}\)
= \(-2\sqrt{2}\)