Find-the-value-of-x-2-3-x-2-3-x-4-

Question Number 7251 by Tawakalitu. last updated on 19/Aug/16

F i n d t h e v a l u e o f x

{(\sqrt{2 + \sqrt{3}})}^{x} + {(\sqrt{2 - \sqrt{3}})}^{x} = 4

Commented by sou1618 last updated on 19/Aug/16

(*) \cdot \cdot \cdot {(2 + \sqrt{3})}^{x / 2} + {(2 - \sqrt{3})}^{x / 2} = 4

\frac{1}{2 + \sqrt{3}} = \frac{1}{2 + \sqrt{3}} \times \frac{2 - \sqrt{3}}{2 - \sqrt{3}} = 2 - \sqrt{3}

(*) \Rightarrow {(2 + \sqrt{3})}^{x / 2} + {(\frac{1}{2 + \sqrt{3}})}^{x / 2} = 4

s e t y = {(2 + \sqrt{3})}^{x / 2} (y \neq 0)

\Rightarrow y + \frac{1}{y} = 4 (y \neq 0)

\Rightarrow y^{2} - 4 y + 1 = 0

\Rightarrow y = 2 \pm \sqrt{4 - 1}

= 2 \pm \sqrt{3}

{(2 + \sqrt{3})}^{x / 2} = 2 + \sqrt{3}, 2 - \sqrt{3}

{(2 + \sqrt{3})}^{x / 2} = 2 + \sqrt{3}, \frac{1}{2 + \sqrt{3}}

\frac{x}{2} = 1, - 1

x = \pm 2

Commented by Rasheed Soomro last updated on 19/Aug/16

N i c e!

Commented by Tawakalitu. last updated on 19/Aug/16

W o w, t h a n k s s o m u c h s i r, G o d b l e s s y o u .

Commented by peter james last updated on 19/Aug/16

v e r y n i c e \dots