Question

If $x=5-2\sqrt{6}$. Find the value of $x-\frac{1}{x}$.

Answer 1

Given, $x=5-2\sqrt{6}$.

$\therefore \frac{1}{x}=\frac{1}{5-2\sqrt{6}}=\frac{1}{5-2\sqrt{6}}\times \frac{5+2\sqrt{6}}{5+2\sqrt{6}}=\frac{5+2\sqrt{6}}{25-24}=5+2\sqrt{6}$.

$\therefore x-\frac{1}{x}=5-2\sqrt{6}-(5+2\sqrt{6})=5-2\sqrt{6}-5-2\sqrt{6}=4\sqrt{6}$.

$x+\frac{1}{x}=5-2\sqrt{6}+(5+2\sqrt{6})=5-2\sqrt{6}+5+2\sqrt{6}=10$.

Therefore, the required value is $10$.