How do you solve for x: in 2^x + 2^-x = 5/2?

May 14, 2016

$x = \pm 1$

Explanation:

(I'm hoping that someone will come up with a better method of solving, since the following is simply "solution by observation").

We are told: ${2}^{x} + {2}^{- x} = \frac{5}{2}$

Observe that
$\textcolor{w h i t e}{\text{XXXX}} \frac{5}{2} = \frac{4}{2} + \frac{1}{2}$

$\textcolor{w h i t e}{\text{XXXXX}} = 2 + \frac{1}{2}$

$\textcolor{w h i t e}{\text{XXXXX}} = {2}^{1} + \frac{1}{2} ^ 1$

$\textcolor{w h i t e}{\text{XXXXX}} = {2}^{1} + {2}^{- 1}$

So ${2}^{x} + {2}^{- x} = {2}^{1} + {2}^{- 1} = {2}^{-} 1 + {2}^{1}$
which implies

$\textcolor{w h i t e}{\text{XXX}} x = \pm 1$

Graph

Another possible way is looking at a graph