If p = \frac{\sqrt{5}-2}{\sqrt{5}+2} and q = \frac{\sqrt{5}+2}{\sqrt{5}-2}, then what is \left(\frac{p}{q} + \frac{q}{p}\right) equal to?

Explanation

First, p \times q = 1. The expression is \frac{p^2+q^2}{pq} = p^2 + q^2 = (p+q)^2 - 2pq. Calculating p+q = \frac{(\sqrt{5}-2)^2 + (\sqrt{5}+2)^2}{5-4} = \frac{2(5+4)}{1} = 18. Therefore, p^2+q^2 = 18^2 - 2(1) = 324 - 2 = 322.

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