Question

√7-2/√7+2=a√7+b
please solve this question​

Answer 1

$\large\underline{\sf{Solution-}}$

It is provided that,

$\sf \: \dfrac{ \sqrt{7} - 2}{ \sqrt{7} + 2 } = a \sqrt{7} + b$

On rationalizing the denominator,

$\sf \: \dfrac{ \sqrt{7} - 2}{ \sqrt{7} + 2 } \times \dfrac{ \sqrt{7} - 2}{ \sqrt{7} - 2} = a \sqrt{7} + b$

$\sf \: \dfrac{ (\sqrt{7} - 2)^{2} }{ (\sqrt{7})^{2} - {2}^{2} } = a \sqrt{7} + b$

We know that

$\sf \: {(x - y)}^{2} = {x}^{2} + {y}^{2} - 2xy$

and

$\sf \: (x + y)(x - y) = {x}^{2} - {y}^{2}$

Therefore, we have

$\sf \: \dfrac{ 7 + 4 - 4 \sqrt{7} }{ 7 - 4 } = a \sqrt{7} + b$

$\sf \: \dfrac{11 - 4 \sqrt{7} }{3} = a \sqrt{7} + b$

$\sf \:\dfrac{11}{3} - \dfrac{4 \sqrt{7} }{3} = a \sqrt{7} + b$

So, if we compare,

$\sf \: b = \dfrac{11}{3}$

$\sf \: a = - \: \dfrac{4}{3}$