Question

If $a=2+\sqrt{5}$, then find the value of $a-\frac{1}{a}$

Answer 1

$\frac{1}{a}=\frac{1}{2+\sqrt{5}}=\frac{1}{2+\sqrt{5}}\ast \frac{2-\sqrt{5}}{2-\sqrt{5}}=\frac{2-\sqrt{5}}{4-5}=\sqrt{5}-2$

Hence, $a-\frac{1}{a}=2+\sqrt{5}-\sqrt{5}+2=4$