Finding the value of $2^{3x-1}$

Question

Given $$2^{2x-1}-2^x = 12,$$ find the value of $2^{3x-1}$.

Simpiflying the equation:

$$2^{x}.2^{x}.2^{-1}-2^x = 2^2 \cdot 3$$

This is where I'm stuck.

Regards!

Answer 1

Guide:

Let $y = 2^x>0$, then we have $$\frac12 y^2 - y = 12$$

which is just a quadratic equation. Get the positive root.

$$2^{3x-1}=\frac12(2^x)^3=\frac{y^3}2$$