Question

If $x=5-4\sqrt{6},$ then, $x+\frac{1}{x}$ is $\frac{350}{71}-\frac{288}{71}\sqrt{m}$, where $m$ is

Answer 1

$x=5-4\sqrt{6}\frac{1}{x}=\frac{1}{5-4\sqrt{6}}Werationalizethedenominator,\frac{1}{x}=\frac{1}{5-4\sqrt{6}}\ast \frac{5+4\sqrt{6}}{5+4\sqrt{6}}=\frac{5+4\sqrt{6}}{25-16\ast 6}=-\left(\frac{5+4\sqrt{6}}{71}\right)x-\frac{1}{x}=5-4\sqrt{6}-\left(\frac{5+4\sqrt{6}}{71}\right)=\frac{355-5-(284+4)\sqrt{6}}{71}=\frac{350-288\sqrt{6}}{71}$\\ \\