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

Question Number 156338 by MathSh last updated on 10/Oct/21

Solve in R

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

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

x > 1

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

g (x) = \frac{2}{x} - 1

b o t h f u n c t i o n s a r e d e c r e a s i n g .

f (1) \to + \infty

g (1) = 1 < f (1)

f (2) = \frac{\sqrt{3}}{3}

f (2) = 0 < f (2)

f o r x > 2 :

f (x) > 0

g (x) < 0

t h a t m e a n s f o r x > 1 : f (x) > g (x)

\Rightarrow n o s o l u t i o n f o r f (x) = g (x)!

Commented by MathSh last updated on 10/Oct/21

Perfect my dear Ser, thank you

Answered by peter frank last updated on 10/Oct/21

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

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

x^{2} = {(2 - x)}^{2} (x^{2} - 1)

x^{2} = [4 - 4 x + x^{2}] (x^{2} - 1)

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

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

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

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

\dots . .