Question

If $p=5+2\sqrt{6},x=\frac{1}{p}$. Find the value of $p^2+x^2$.

Answer 1

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

and also $x=\frac{1}{p}=\frac{1}{5+2\sqrt{6}}=\frac{5-2\sqrt{6}}{5^2-(2\sqrt{6}{)}^2}=\frac{5-2\sqrt{6}}{25-24}=5-2\sqrt{6}$

$p^2+x^2=(5+2\sqrt{6}{)}^2+(5-2\sqrt{6}{)}^2=2(5^2+(2\sqrt{6}{)}^2)=2(49)=98$