Question

If $x=5-4\sqrt{6}$, $x+\frac{1}{x}$ is equal to $\frac{350}{71}-\frac{288}{71}\sqrt{m}$.
Find the value of m.

Answer 1

$x=5-4\sqrt{6}$ and $\frac{1}{x}=\frac{1}{5-4\sqrt{6}}$.

On rationalizing the denominator,

$\frac{1}{x}=\frac{1}{5-4\sqrt{6}}\times \frac{5+4\sqrt{6}}{5+4\sqrt{6}}$

$=\frac{5+4\sqrt{6}}{25-16\ast 6}$ $=-\left(\frac{5+4\sqrt{6}}{71}\right)$.

Now, $x-\frac{1}{x}=5-4\sqrt{6}-\left(\frac{5+4\sqrt{6}}{71}\right)$

$=\frac{355-5-(284+4)\sqrt{6}}{71}$

$=\frac{350-288\sqrt{6}}{71}$.

$\therefore$ Value of $m=6$.