Question

$x=\frac{\sqrt{5}-2}{\sqrt{5}+2}$ and $y=\frac{\sqrt{5}+2}{\sqrt{5}-2}$.

If $x^2$ is $161-72\sqrt{m}$, then value of $m$ is ?

Answer 1

Given, $x^2$ is $161-72\sqrt{m}$.............(1)

Rationalize the denominator,

$x=\frac{\sqrt{5}-2}{\sqrt{5}+2}\times \frac{\sqrt{5}-2}{\sqrt{5}-2}$

$x=\frac{5+4-4\sqrt{5}}{5-4}$

$x=\frac{9-4\sqrt{5}}{1}$.

$x^2=(9-4\sqrt{5})(9-4\sqrt{5})$

$=81+16\times 5-72\sqrt{5}$

$=161-72\sqrt{5}$ ....................(2)

Comparing equations (1) and (2), we get,

$m=5$.