Question

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

Answer 1

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

$\frac{49+25+14\sqrt{5}}{44}-\frac{(49+25-14\sqrt{5})}{44}$

$=\frac{14\sqrt{5}+14\sqrt{5}}{44}=\frac{2\times 14\sqrt{5}}{44}=\frac{7\sqrt{15}}{11}$

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

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