Question

If $p=\frac{(2-\sqrt{5})}{(2+\sqrt{5})}$ and $q=\frac{(2+\sqrt{5})}{(2-\sqrt{5})}$, find the values of $p-q$

Answer 1

$p-q=\frac{(2-\sqrt{5})}{(2+\sqrt{5})}–\frac{(2+\sqrt{5})}{(2-\sqrt{5})}$

So by rationalizing the denominator, we get

$=\frac{\left[\right(2–\sqrt{5}{)}^2–(2+\sqrt{5}{)}^2]}{[2^2–(\sqrt{5}{)}^2]}$

$=\frac{[4+5–4\sqrt{5}–(4+5+4\sqrt{5}\left)\right]}{[4–5]}$

$=\frac{[9–4\sqrt{5}–9–4\sqrt{5}]}{-1}$

$=\frac{[-8\surd 5]}{-1}$

$=8\surd 5$