How to approach this

Question

$2^{33x-2} + 2^{11x+2} = 2^{22x+1} +1$

The question is to find the sum of all the solutions for $x$.

How does one approach these type of problems? I tried to take $2^{11x}$ as $n$, but failed to solve this. Help me out please.

Answer 1

Multiply by 4 to get cubic $$ t^3 - 8 t^2 + 16 t -4$$

The three roots of the cubic polynomial are all positive. If we call the correct $x$ values $a,b,c$ we find the product $$ 2^{11a} 2^{11b} 2^{11c} = 4 \; , $$ $$ 2^{11a+11b+11c} = 4 $$ $$ 11(a+b+c) = 2 $$

Answer 2

Write $$(2^{11x})^3\cdot \frac{1}{4}+2^{11x}\cdot 4=(2^{11x})^2\cdot 2+1$$ and now substitute $$2^{11x}=t$$