Question

Find the values of $a$ and $b$ in $\frac{3-\sqrt{5}}{3+2\sqrt{5}}$ $=a\sqrt{5}-\frac{b}{11}$.

Answer 1

Now,

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

$=\frac{(3-\sqrt{5})(3-2\sqrt{5})}{3^2-(2\sqrt{5}{)}^2}$

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

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

According to the problem,

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

So, comparing both sides we get,

$a=\frac{9}{11}$ and $b=19$.