Question

If $x=5-4\sqrt{6},$ then:

If $x-\frac{1}{x}$ is $\frac{360}{71}-\frac{280}{71}\sqrt{m}$, then $m$ is

Answer 1

$x=5-4\sqrt{6}$
$\frac{1}{x}=\frac{1}{5-4\sqrt{6}}$
We rationalize the denominator,
$\frac{1}{x}=\frac{1}{5-4\sqrt{6}}\times \frac{5+4\sqrt{6}}{5+4\sqrt{6}}$
$=\frac{5+4\sqrt{6}}{25-16\times 6}$
$=-\left(\frac{5+4\sqrt{6}}{71}\right)$
$x-\frac{1}{x}=5-4\sqrt{6}+\left(\frac{5+4\sqrt{6}}{71}\right)$
$=\frac{355+5-(284-4)\sqrt{6}}{71}$
$=\frac{360-280\sqrt{6}}{71}$
$\therefore m=6$