Question

if x=3+√2 then find the value of x-1/x

Answer 1

Answer

18 + 8√2 / 7

Given

x = 3 + √2

To Find

Value of x - 1/x

Solution

$\rm x = 3+\sqrt{2}\\\\\to \rm \dfrac{1}{x}=\dfrac{1}{3+\sqrt{3}}$

Rationalize the denominator

$\to \rm \dfrac{1}{x}=\dfrac{1}{3+\sqrt{2}}\times \dfrac{3-\sqrt{2}}{3+\sqrt{2}}\\\\\to \rm \dfrac{1}{x}=\dfrac{3-\sqrt{2}}{(3)^2-(\sqrt{2})^2}\\\\\to \rm \dfrac{1}{x}=\dfrac{3-\sqrt{2}}{9-2}\\\\\to \rm \dfrac{1}{x}=\dfrac{3-\sqrt{2}}{7}$

Now , Our required value ,

$\rm x-\dfrac{1}{x}\\\\\to \rm (3+\sqrt{2})-\dfrac{3-\sqrt{2}}{7}\\\\\to \rm \dfrac{7(3+\sqrt{2})-(3-\sqrt{2})}{7}\\\\\to \rm \dfrac{21+7\sqrt{2}-3+\sqrt{2}}{7}\\\\\to \rm \dfrac{18+8\sqrt{2}}{7}$