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 have found out that the answers were 0,1, and -1. But I used sort of a guess-and check way.

$2(2^x-1)x^2+(2^{x^2}-2)x=2^{x+1}-2$

I expanded it into:

$(x^2-1)(2^{x+1}-2)+x(2^{x^2}-2)=0$

I just found all the possibilities to make each group of equation 0 resulting in 0,1, and -1.

How can I prove this is correct instead of trial/error?

Answer 1

Mistake when you divide by 2. You get: $$2^{x^2-1} = 2^{(x+1)(x-1)} = 2^{{x+1}^{x-1}}$$

Now maybe it gets easier?