How to solve equation with unknown nth degree

Question

How it is possible to solve this equation

$$3*2^\sqrt x+2^{3-\sqrt x}=25$$

I have used formula $x^y=y^x$

But seems that it is not right answer. I just don't know hot to get rid of $\sqrt x$ exponent.

Answer 1

hint

Put $$2^\sqrt {x}=z. $$

it becomes

$$3z+\frac {8}{z}=25$$ or

$$3z^2-25z+8=0$$

with $\Delta=625-96=(23)^2$.

Answer 2

Multiply by $2^{\sqrt x}$ and rewrite as

$$3\cdot\left(2^{\sqrt x}\right)^2-25\cdot2^{\sqrt x}+2^{3}=0.$$

Then

$$2^\sqrt{x}=\frac{25\pm23}{6}=\frac13\text{ or }8$$

and $$x=\left(\log_2\frac13\right)^2\text{ or }9.$$