Solve-in-R-1-x-2-1-1-2-x-

Question Number 156361 by mr W last updated on 10/Oct/21

Solve in R

\frac{1}{\sqrt{x^{2} - 1}} = 1 - \frac{2}{x}

Commented by cortano last updated on 11/Oct/21

Commented by mr W last updated on 11/Oct/21

x = - \frac{1}{\cos (\frac{1}{2} \sin^{- 1} \frac{\sqrt{5} - 1}{2})}

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

Answered by MJS_new last updated on 10/Oct/21

\sqrt{x^{2} - 1} \neq 0 \land x^{2} - 1 > 0 \land x \neq 0 \land 1 - \frac{2}{x} > 0

\Leftrightarrow

x < - 1 \lor - 1 < x < 0 \lor x > 2

squaring & transforming \Rightarrow

x^{4} - 4 x^{3} + 2 x^{2} + 4 x - 4 = 0

x = t + 1

t^{4} - 4 t^{2} - 1 = 0

(t^{2} - 2 - \sqrt{5}) (t^{2} - 2 + \sqrt{5}) = 0

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

x = 1 \pm \sqrt{2 + \sqrt{5}}

both solve the given equation

Commented by mr W last updated on 10/Oct/21

t h a n k s s i r!

Commented by MJS_new last updated on 10/Oct/21

it was just a typo in the first line which I

inserted later

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