Question

find the value of a and b :
$\frac{7 + \sqrt{5} }{7 - \sqrt{5} } - \frac{7 - \sqrt{5} }{7 + \sqrt{5} } = a + \frac{7 \sqrt{5} }{11} b$
​

Answer 1

$\mathsf{Question :\;\dfrac{7 + \sqrt{5}}{7 - \sqrt{5}} - \dfrac{7 - \sqrt{5}}{7 + \sqrt{5}}}$

Taking LCM :

$\mathsf{\implies \dfrac{(7 + \sqrt{5})(7 + \sqrt{5}) - (7 - \sqrt{5})(7 - \sqrt{5})}{(7 - \sqrt{5})(7 + \sqrt{5})}}$

$\mathsf{\implies \dfrac{(7 + \sqrt{5})^2 - (7 - \sqrt{5})^2}{(7 - \sqrt{5})(7 + \sqrt{5})}}$

$\mathsf{\implies \dfrac{(7)^2 + (\sqrt{5})^2 + 2(7)\sqrt{5} - [(7)^2 +(\sqrt{5})^2 - 2(7)\sqrt{5}]}{(7 - \sqrt{5})(7 + \sqrt{5})}}$

$\mathsf{\implies \dfrac{(7)^2 + (\sqrt{5})^2 + 2(7)\sqrt{5} - (7)^2 - (\sqrt{5})^2 + 2(7)\sqrt{5}}{(7 - \sqrt{5})(7 + \sqrt{5})}}$

$\mathsf{\implies \dfrac{4(7)\sqrt{5}}{(7 -\sqrt{5})(7 + \sqrt{5})}}$

$\mathsf{\implies \dfrac{28\sqrt{5}}{(7)^2 - (\sqrt{5})^2}}$

$\mathsf{\implies \dfrac{28\sqrt{5}}{49 - 5}}$

$\mathsf{\implies \dfrac{28\sqrt{5}}{44}}$

$\mathsf{\implies \dfrac{7\sqrt{5}}{11}}$

$\mathsf{\implies 0 + \dfrac{7\sqrt{5}}{11} = a + \dfrac{7\sqrt{5}}{11}b}$

Comparing both sides :

●  a = 0

●  b = 1