Please help to find the solution

Question

$$2^{x-2} + 2^{3-x} = 3$$ Please find the value of $x$ and give the solution process.

Answer 1

$2^{x-2} + 2^{3-x}= 3 $ $ \iff $

$ 2^{x} * \frac{1}{4} + 8*2^{-x} = 3 $ $ \iff (2^{x})^{2} * \frac{1}{4}-3*2^{-x} + 8 = 0 $ we put $2^{x}=y$ we have, $ \frac{1}{4}y²-3y+8=0 \iff y=4 $ or $ y=8 $ now you can find easily that $ x=2 $ and $ x=3 $

Answer 2

HINT:

$$3=\dfrac y{2^2}+\dfrac{2^3}y$$ where $2^x=y$

Now multiply both sides by $y(\ne0)$ for finite $x$