Question

if x=1-√2, find the value of x-1/x​

Answer 1

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Question :

$\sf \small If \: x = 1 - \sqrt{2} \: then \: find \: the \: value \: of \: x - \frac{1}{x}$

To find :

$\sf \small The \: value \: of \: x - \frac{1}{x}$

Solution :

x → 1 – √2

So 1/x must be 1/1 - √2

So we need to rationalise 1/x

$\frac{1}{x} \rarr \frac{1}{1 - \sqrt{2} }$

$\frac{1}{x} = \frac{1(1 + \sqrt{2}) }{(1 - \sqrt{2})(1 + \sqrt{2}) }$

$\frac{1}{x} = \frac{1 + \sqrt{2} }{ {(1)}^{2} - {( \sqrt{2}) }^{2} }$

$\frac{1}{x} = \frac{1 + \sqrt{2} }{ 1 - 2}$

$\frac{1}{x} = \frac{1 + \sqrt{2} }{ - 1}$

$\frac{1}{x} = - 1 + \sqrt{2}$

Now we need to solve x - 1/x

$x - \frac{1}{x}$

$\rarr1 - \sqrt{2} - ( - 1 + \sqrt{2} )$

$\rarr1 - \sqrt{2} + 1 - \sqrt{2}$

$\rarr1 + 1$

$\rarr2$

$\sf \small So \: the \: value \: of \: x - \frac{1}{x} \: is \: \boxed{2}$

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