Find all real solutions for $x$ in $ 2(2^x- 1) x^2 + (2^{x^2}-2)x = 2^{x+1} -2 . $

Question

Find all real solutions for $x$ in $$ 2(2^x- 1) x^2 + (2^{x^2}-2)x = 2^{x+1} -2 . $$

I noticed that $$2(2^x- 1) x^2 + (2^{x^2}-2)x = 2^{x+1} -2=2(2^x−1)(x^2-1)+(2^{x^2}−2)x=0.$$

I know that $2^{n+1}-1$ will always be greater than $2^n-1$ for all positive integers. $2^{n-1}-1$ will always be less than $2^n-1$.

How do I solve this problem?

Answer 1

$1$, $-1$ and $0$ are roots.

Let $x\notin\{0,1,-1\}$.

Thus, $$2(2^x-1)x^2+2x(2^{x^2-1}-1)=2(2^x-1)$$ or $$(2^x-1)(x^2-1)+(2^{x^2-1}-1)x=0$$ or $$\frac{2^x-1}{x}+\frac{2^{x^2-1}-1}{x^2-1}=0,$$ which has no real roots because for any $a\neq0$ we have $$\frac{2^a-1}{a}>0.$$