Decode the equation to quadratic equation

Question

Tring to create a equation of the form $$2^{x^2} : 2^x = 8:1$$ into the form of $$ax^2 + bx + c $$ or find the value of $x$.

The solution to this is $$x = \frac{1\pm\sqrt{13}}{2}$$

Answer 1

You can rewrite $$a:b=c:d$$

as $$\frac ab=\frac cd$$

and use $\frac{a^x}{a^y} = a^{x-y}$ and $8=2^3$

Answer 2

$2^{x^2} : 2^x = 8:1 \iff 2^{x^2-x}=8 \iff x^2-x=3$.