We need to solve \(2(2^x-1)x^2+(2^{x^2}-2)x-2^{x+1}-2=0\).

Let \(f(x) = 2(2^x-1)x^2+(2^{x^2}-2)x-2^{x+1}-2\).

If you graph f(x) using any method, you find that it only touches x-axis at one point.

Graph link: https://www.desmos.com/calculator/wxie170yvi

We will use Newton-Rhapson to find solutions.

\(f'(x) = 4x(2^x-1)+2x^22^x\ln2+2^{x^2}-2+2x^2(2^{x^2}\ln2)-2^{x+1}\ln2\)

As this is too complicated, I solved it with a program.

Here is the source code in C++:

#include

using namespace std;

double f(double x){

return 2*(pow(2,x)-1)*x*x+(pow(2,x*x)-2)*x-pow(2,x+1)-2;

}

double f_prime(double x){

return 4*x*(pow(2,x)-1)+2*x*x*pow(2,x)*log(2)+pow(2,x*x)-2+2*x*x*(pow(2,x*x))*log(2)-pow(2,x+1)*log(2);

}

int main(){

double x = 1.1;

for (int i=0;i<1000;i++){ //iterate 1000 times

x -= f(x)/f_prime(x);

}

printf("Solution: %100g", x);

}

Output: Solution: 1.32006