Question

If $\frac{\sqrt{7}-1}{\sqrt{7}+1}-\frac{\sqrt{7}+1}{\sqrt{7}-1}=a+b\sqrt{7}$, then find the value of $a$ and $b$.

Answer 1

$\frac{\sqrt{7}-1}{\sqrt{7}+1}-\frac{\sqrt{7}+1}{\sqrt{7}-1}=a+b\sqrt{7}$

$\Rightarrow \frac{(\sqrt{7}-1{)}^2-(\sqrt{7}+1{)}^2}{(\sqrt{7}+1)(\sqrt{7}-1)}=a+b\sqrt{7}$

$\Rightarrow \frac{7+1-2\sqrt{7}-7-1-2\sqrt{7}}{(\sqrt{7}{)}^2-(1{)}^2}$$=a+b\sqrt{7}$

$\Rightarrow \frac{-4\sqrt{7}}{6}=a+b\sqrt{7}$

$\Rightarrow 0-\frac{2}{3}\sqrt{7}=a+b\sqrt{7}$

$\Rightarrow a=0,b=-\frac{2}{3}$.