Question

x-1/x and x2-1/x2 if x+1/x=√5

Answer 1

Solution of 1st one is below

Answer 2

Ans.

${x + \frac{1}{x} } = \sqrt{5 } \\ or \: \: {(x + \frac{1}{x}) }^{2} = ( { \sqrt{5}) }^{2} \\ or \: \: {(x - \frac{1}{x}) }^{2} + 4.x. \frac{1}{x} = 5 \\ or \: \: {(x - \frac{1}{x}) }^{2} + 4 = 5 \\ or \: \: {(x - \frac{1}{x}) }^{2} \: = 1 \\ \\ then \: x - \frac{1}{x} = 1 \: \: \: or \: \: \: x - \frac{1}{x} = - 1$

When
$x - \frac{1}{x} = 1 \: \: then \: \: \\ {x}^{2} - \frac{1}{ {x}^{2} } \\ = (x + \frac{1}{x} )(x - \frac{1}{x}) \\ = \sqrt{5} \times 1 \\ = \sqrt{5}$

and when
$x - \frac{1}{x} = - 1 \: \: \: then \\ {x}^{2} - \frac{ {1} }{ {x}^{2} } \\ = (x + \frac{1}{x} )(x - \frac{1}{x} ) \\ = \sqrt{5} \times ( - 1) \\ = - \sqrt{5}$


I HOPE THAT THIS HELPS YOU.