x-1-x-1-5-2-x-

Question Number 186241 by a.lgnaoui last updated on 02/Feb/23

x + \frac{1}{x} = \frac{- 1 + \sqrt{5}}{2}

x ?

Commented by MJS_new last updated on 02/Feb/23

are you unable to solve a simple quadratic ?

Commented by a.lgnaoui last updated on 02/Feb/23

T h e v a l u e {(5 x)}^{5} = ?, w i t h o u t c a l c u l o f x .

Commented by a.lgnaoui last updated on 02/Feb/23

I w a n t b y t h i s s i m p l e e q u a t i o n t o

e v a l u a t e {(5 x)}^{5} w i t h o u t c a l c u l o f x ?

Answered by Frix last updated on 02/Feb/23

x = \frac{- 1 + \sqrt{5}}{4} \pm \frac{\sqrt{10 + 2 \sqrt{5}}}{4} i

Answered by MJS_new last updated on 02/Feb/23

x + \frac{1}{x} = \frac{- 1 + \sqrt{5}}{2}

\Leftrightarrow

x^{2} + (\frac{1 - \sqrt{5}}{2}) x + 1 = 0

we know

(x^{2} + (\frac{1 - \sqrt{5}}{2}) x + 1) (x^{2} + (\frac{1 + \sqrt{5}}{2}) x + 1) (x - 1) = x^{5} - 1 = 0

\Rightarrow x^{5} = 1 \Rightarrow {(5 x)}^{5} = 3125

Commented by a.lgnaoui last updated on 02/Feb/23

g o o d