Question

√7-√5/√7+√5=a+b√35, then find the value of a^2+4ab+b^2​

Answer 1

Answer:

value

Step-by-step explanation:

√7-√5/√5=a+b√35

Answer 2

Answer:

The value of a² + 4ab + b² is 13.

Step-by-step-explanation:

We have given that,

$\displaystyle{\sf\:\dfrac{\sqrt{7}\:-\:\sqrt{5}}{\sqrt{7}\:+\:\sqrt{5}}\:=\:a\:+\:b\:\sqrt{35}}$

We have to find the value of a² + 4ab + b².

Now,

$\displaystyle{\sf\:\dfrac{\sqrt{7}\:-\:\sqrt{5}}{\sqrt{7}\:+\:\sqrt{5}}\:=\:a\:+\:b\:\sqrt{35}}$

$\displaystyle{\implies\sf\:\dfrac{\sqrt{7}\:-\:\sqrt{5}}{\sqrt{7}\:+\:\sqrt{5}}\:\times\:\dfrac{\sqrt{7}\:-\:\sqrt{5}}{\sqrt{7}\:-\:\sqrt{5}}\:=\:a\:+\:b\:\sqrt{35}\;\quad\:\dots\:[\:Rationalising\:the\:denominator\:]}$

$\displaystyle{\implies\sf\:\dfrac{(\:\sqrt{7}\:-\:\sqrt{5}\:)\:(\:\sqrt{7}\:-\:\sqrt{5}\:)}{(\:\sqrt{7}\:+\:\sqrt{5}\:)\:(\:\sqrt{7}\:-\:\sqrt{5}\:)}\:=\:a\:+\:b\:\sqrt{35}}$

$\displaystyle{\implies\sf\:\dfrac{(\:\sqrt{7}\:-\:\sqrt{5}\:)^2}{(\:\sqrt{7}\:)^2\:-\:(\:\sqrt{5}\:)^2}\:=\:a\:+\:b\:\sqrt{35}\:\quad\:\dots\:[\:(\:a\:+\:b\:)\:(\:a\:-\:b\:)\:=\:a^2\:-\:b^2\:]}$

$\displaystyle{\implies\sf\:\dfrac{(\:\sqrt{7}\:)^2\:-\:2\:\times\:\sqrt{7}\:\times\:\sqrt{5}\:+\:(\:\sqrt{5}\:)^2}{7\:-\:5}\:=\:a\:+\:b\:\sqrt{35}\:\quad\;\dots\:[\:(\:a\:-\:b\:)^2\:=\:a^2\:-\:2ab\:+\:b^2\:]}$

$\displaystyle{\implies\sf\:\dfrac{7\:-\:2\:\sqrt{35}\:+\:5}{2}\:=\:a\:+\:b\:\sqrt{35}}$

$\displaystyle{\implies\sf\:\dfrac{7\:+\:5\:-\:2\:\sqrt{35}}{2}\:=\:a\:+\:b\:\sqrt{35}}$

$\displaystyle{\implies\sf\:\dfrac{12\:-\:2\:\sqrt{35}}{2}\:=\:a\:+\:b\:\sqrt{35}}$

$\displaystyle{\implies\sf\:\cancel{\dfrac{12}{2}}\:+\:\dfrac{-\:\cancel{2}}{\cancel{2}}\:\sqrt{35}\:=\:a\:+\:b\:\sqrt{35}}$

$\displaystyle{\implies\boxed{\pink{\sf\:a\:=\:6}}\sf\quad\:\&\:\quad\:\boxed{\pink{\sf\:b\:=\:-\:1}}\sf\:\quad\:\dots\:[\:By\:comparing\:]}$

Now, we have to find the value of

$\displaystyle{\sf\:a^2\:+\:4ab\:+\:b^2}$

$\displaystyle{\implies\sf\:(\:6\:)^2\:+\:4\:\times\:6\:\times\:(\:-\:1\:)\:+\:(\:-\:1\:)^2}$

$\displaystyle{\implies\sf\:36\:-\:24\:+\:1}$

$\displaystyle{\implies\sf\:12\:+\:1}$

$\displaystyle{\implies\sf\:13}$

$\displaystyle{\therefore\:\underline{\boxed{\red{\sf\:a^2\:+\:4ab\:+\:b^2\:=\:13\:}}}}$