Question

if x=
$\sqrt{2} - 1$
Then find
$x - \frac{1}{x}$
​

Answer 1

$x - \frac{1}{x} \\ = ( \sqrt{2} - 1) - \frac{1}{ (\sqrt{2 } - 1) } \\ = \frac{( \sqrt{2} - 1) {}^{2} - 1 }{( \sqrt{2} - 1)} \\ = \frac{2 - 2 \times \sqrt{2} \times 1 + 1 {}^{2} - 1 }{( \sqrt{2} - 1) } \\ = \frac{2 - 2 \sqrt{2} + 1 - 1}{ \sqrt{2} - 1} \\ = \frac{3 - 2 \sqrt{2} - 1 }{ \sqrt{2} - 1 } \\ = \frac{2 - 2 \sqrt{2} }{ \sqrt{2 } - 1} \\ = \frac{2 - 2 \sqrt{2} }{ \sqrt{2 } - 1} \times \frac{ \sqrt{2} + 1 }{ \sqrt{2} + 1 } \\ = \frac{(2 - 2 \sqrt{2})( \sqrt{2} + 1) }{2 - 1} \\ = 2( \sqrt{2} + 1) - 2 \sqrt{2} ( \sqrt{2} + 1) \\ = 2 \sqrt{2} + 2 - 4 - 2 \sqrt{2} \\ = 2 - 4 \\ = - 2 \: ans...$

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