Question

Determine rational number p and q if
$\frac{l-\sqrt{5}}{l-\sqrt{5}}-\frac{7-\sqrt{5}}{7+\sqrt{5}}=p-7\sqrt{5}q$

Answer 1

$\to L.H.S:$ Rationalising the denominators:

$\Rightarrow \frac{7+\sqrt{5}}{7-\sqrt{5}}\times \frac{7+\sqrt{5}}{7+\sqrt{5}}-\frac{7-\sqrt{5}}{7+\sqrt{5}}\times \frac{7-\sqrt{5}}{7-\sqrt{5}}$

$\Rightarrow \frac{(7+\sqrt{5}{)}^2}{(7{)}^2-(\sqrt{5}{)}^2}-\frac{(7-\sqrt{5}{)}^2}{(7{)}^2-(\sqrt{5}{)}^2}$

$\Rightarrow \frac{\left[\right(7+\sqrt{5})-(7-\sqrt{5}\left)\right]\left[\right(7+\sqrt{5})+(7-\sqrt{5}\left)\right]}{49-5}$

$\Rightarrow \frac{\left(2\sqrt{5}\right)\left(14\right)}{44}\Rightarrow \frac{7}{11}\sqrt{5}$

$\to$ Comparing the rational and irrational terms on both $L.H.S$ and $R.H.S$, we get,

$p=0$ and $-7\sqrt{5}q=\frac{7}{11}\sqrt{5}$

$\Rightarrow p=0$ and $q=\frac{-1}{11}$