Question

If
$x + \frac{1}{x}$

equals to 10, then what's the answer of
$x - \frac{1}{x}$
​

Answer 1

$±4 \sqrt{6}$

Step-by-step explanation:

$Given, \: \: x + \frac{1}{x} = 10 \\ square \: \: on \: \: both \: \: sides, \\ (x + \frac{1}{x} )^{2} = {10}^{2} \\ {x}^{2} + \frac{1}{ {x}^{2} } + 2 \times x \times \frac{1}{x} = 100 \\ {x}^{2} + \frac{1}{ {x}^{2} } + 2 = 100 \\ \boxed{ {x}^{2} + \frac{1}{ {x}^{2} } = 98} \\ Now, \: \: \: (x - \frac{1}{x} )^{2} = {x}^{2} + \frac{1}{ {x}^{2} } - 2 \times x \times \frac{1}{x} \\ (x - \frac{1}{x} )^{2} = \boxed{ {x}^{2} + \frac{1}{ {x}^{2} }} - 2 \\ (x - \frac{1}{x} )^{2} = 98 - 2 \\ (x - \frac{1}{x} )^{2} = 96 \\ x - \frac{1}{x} = ± \sqrt{96} \\\boxed{ x - \frac{1}{x} = ±4 \sqrt{6}}$