Question

please answer my question ​

Answer 1

EXPLANATION.

$\implies \dfrac{7 + \sqrt{5} }{7 - \sqrt{5} } - \dfrac{7 - \sqrt{5} }{7 + \sqrt{5} } = p - 7\sqrt{5} q$

As we know that,

Rationalizes the equation one by one, we get.

First we rationalize = (7 + √5)/(7 - √5).

$\implies \dfrac{7 + \sqrt{5} }{7 - \sqrt{5} } \times \dfrac{7 + \sqrt{5} }{7 + \sqrt{5} }$

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

$\implies \dfrac{49 + 5 + 14\sqrt{5} }{49 - 5} = \dfrac{54 + 14\sqrt{5} }{44}$

Now, we rationalizes = (7 - √5)/(7 + √5).

$\implies \dfrac{7 - \sqrt{5} }{7 + \sqrt{5} } \times \dfrac{7 - \sqrt{5} }{7 - \sqrt{5} }$

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

$\implies \dfrac{49 + 5 - 14\sqrt{5} }{49 - 5} = \dfrac{54 - 14\sqrt{5} }{44}$

Now, we can write equation as,

$\implies \dfrac{54 + 14\sqrt{5} }{44} - \dfrac{54 - 14\sqrt{5} }{44}$

$\implies \dfrac{54 + 14 \sqrt{5} - 54 + 14\sqrt{5} }{44} = \dfrac{28\sqrt{5} }{44}$

$\implies \dfrac{28\sqrt{5} }{44} = p - 7\sqrt{5}$

$\implies \dfrac{4 \times 7\sqrt{5} }{44} = p - 7\sqrt{5}$

$\implies \dfrac{7\sqrt{5} }{11} = p - 7\sqrt{5}$

$\implies p = 0 \ \ \ \ q = \dfrac{-1}{11}$