Question

Simplify $\left(\frac{2+\sqrt{5}}{2-\sqrt{5}}\right)+\left(\frac{2-\sqrt{5}}{2+\sqrt{5}}\right)$. [3 Marks]

Answer 1

$\left(\frac{2+\sqrt{5}}{2-\sqrt{5}}\right)+\left(\frac{2-\sqrt{5}}{2+\sqrt{5}}\right)$

It can be solved by rationalizing the denominators individually. For that, we first find the rationalizing factor for both the denominators, ($2-\sqrt{5}$) and ($2+\sqrt{5}$). [1 Mark]

The rationalizing factor for
($2-\sqrt{5}$) would be ($2+\sqrt{5}$) and that for ($2+\sqrt{5}$) would be ($2-\sqrt{5}$).

Rationalizing the given term we get,

$\frac{(2+\sqrt{5})(2+\sqrt{5})}{(2-\sqrt{5})(2+\sqrt{5})}+\frac{(2-\sqrt{5})(2-\sqrt{5})}{(2+\sqrt{5})(2-\sqrt{5})}$ [1 Mark]

$=\frac{4+5+4\sqrt{5}}{4-5}+\frac{4+5-4\sqrt{5}}{4-5}$

$=\frac{4+5+4\sqrt{5}+4+5-4\sqrt{5}}{-1}$
= -18 [1 Mark]