x-1-x-4-then-x-2-1-x-2-

Question Number 165060 by mathlove last updated on 25/Jan/22

x - \frac{1}{x} = 4 t h e n x^{2} - \frac{1}{x^{2}} = ?

Answered by TheSupreme last updated on 25/Jan/22

x^{2} + \frac{1}{x^{2}} = 18

{(x + \frac{1}{x})}^{2} = x^{2} + \frac{1}{x^{2}} + 2 = 20

(x + \frac{1}{x}) = \sqrt{20} = \pm 2 \sqrt{5}

(x - \frac{1}{x}) (x + \frac{1}{x}) = \pm 8 \sqrt{5}

g e n e r a l

(x - \frac{1}{x}) = t

x^{2} + \frac{1}{x^{2}} - 2 = t^{2}

(x^{2} + \frac{1}{x^{2}}) = t^{2} + 2

{(x + \frac{1}{x})}^{2} = t^{2} + 2 + 2 = t^{2} + 4

(x^{2} - \frac{1}{x^{2}}) = \pm t \sqrt{t^{2} + 4}