Equation , powers of two I want to find the sum of the roots of the equation $$4(4^x + 4^{-x}) - 23(2^x + 2^{-x}) + 40 = 0  $$ in real numbers. I tried the substitution $ 2^x = t $ but then it turns into a quartic equation which I couldn't solve. I think its roots sum to zero so I want to prove it without actually finding the roots.

First of all, if $\exists{r}\in\mathbb{R}$ such that $x=r$ is a solution, then $x=-r$ is also a solution.
So the sum of the roots of the equation (if there are any) is obviously $0$.
Second, here is a full analysis of your equation (from this link):

