Question

Find a and b

$\frac{2-\sqrt{5}}{2+3\sqrt{5}}=\sqrt{5}a+b$

Answer 1

$\frac{2-\sqrt{5}}{2+3\sqrt{5}}=\sqrt{5}a+b$

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

$\Rightarrow \frac{4-6\sqrt{5}-2\sqrt{5}+15}{4-45}=\sqrt{5}a+b$

$\Rightarrow \frac{19-8\sqrt{5}}{-41}=\sqrt{5}a+b$

$\Rightarrow \frac{8\sqrt{5}}{41}-\frac{19}{41}=\sqrt{5}a+b$

Comparing both sides , we get

$a=\frac{8}{41}b=-\frac{19}{41}$