Question

If $p=5-2\sqrt{6},$ Find $p^2+\frac{1}{p^2}$

Answer 1

Given $p=5-2\sqrt{6}$

$p+\frac{1}{p}=5-2\sqrt{6}+\frac{1}{5-2\sqrt{6}}=5-2\sqrt{6}+\frac{5-2\sqrt{6}}{5^2-(2\sqrt{6}{)}^2}=5-2\sqrt{6}+5+2\sqrt{6}=10$

$p^2+\frac{1}{p^2}=(p+\frac{1}{p}{)}^2-2=(10{)}^2-2=100-2=98$