Question

If $x=9+4\sqrt{5}$, find the value of $x^2+\frac{1}{x^2}$.

Answer 1

We have,

$x=9+4\sqrt{5}$.

Then,

$\frac{1}{x}$

$=\frac{9-4\sqrt{5}}{81-80}$

$=9-4\sqrt{5}$.

Now,

$x^2+\frac{1}{x^2}$

$={(x+\frac{1}{x})}^2-2$

$={18}^2-2$

$=324-2$

$=322$.