An algebra problem by A Former Brilliant Member

Algebra Level 3

$\Large x+\dfrac{1}{x}=\sqrt{2}$

$\Large x-\dfrac{1}{x}=?$

\sqrt{-2}

Both Neither

\sqrt{-6}

1 solution

Guilherme Niedu
Apr 19, 2016

$x^2 - \sqrt{2}\cdot x + 1 = 0$

$x = \frac{\sqrt{2} \pm i\cdot \sqrt{2}}{2}$

$\frac{1}{x} = \frac{2}{\sqrt{2} \pm i\cdot \sqrt{2}} = \frac{\sqrt{2} ∓ i\cdot \sqrt{2}}{2}$

$x - \frac{1}{x} = i\cdot\sqrt{2} = \sqrt{-2}$