Question

If $\frac{3-\sqrt{5}}{3+2\sqrt{5}}=a\sqrt{5}-\frac{b}{x}$ find $a,b$

Answer 1

$\frac{3-\sqrt{5}}{3+2\sqrt{5}}$

Rationalize the denominator

$\Rightarrow \frac{3-\sqrt{5}}{3+2\sqrt{5}}\times \frac{3-2\sqrt{5}}{3-2\sqrt{5}}$

$\Rightarrow \frac{9-6\sqrt{5}-3\sqrt{5}+10}{3^2-(2\sqrt{5}{)}^2}$

$\Rightarrow \frac{19-9\sqrt{5}}{3^2-(2\sqrt{5}{)}^2}$

$\Rightarrow \frac{19-9\sqrt{5}}{9-20}$

$\Rightarrow \frac{19-9\sqrt{5}}{-11}$

Given that

$\Rightarrow \frac{19-9\sqrt{5}}{-11}=a\sqrt{5}-\frac{b}{x}$

On comparing we get,

$\Rightarrow a=\frac{9}{11};b=19$