Question

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

• A. -18
• B. + 18
• C. 18 + 8√5
• D. 18 - 8√5

Answer 1

The correct option is A

-18

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

can be solved by rationalizing the denominators individually. For that, first find the rationalizing factors for both the denominators, ($2-\sqrt{5}$) and ($2+\sqrt{5}$) 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,

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

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

Since, the denominator is same, i.e, $-1$, we can re-write the equation as

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