Let $A=\{x \in\mathbb{R} \mid x^2+3x-9+2^{x^2+3x-2}+\log_2{(x^2+3x-2)}=0\}$

Question

Find $A=\{x \in\mathbb{R} \mid x^2+3x-9+2^{x^2+3x-2}+\log_2{(x^2+3x-2)}=0\}$.

My attempt:

Let $u=x^2+3x-2\implies u + 2^u + \log_2{u}=7$ and now we have an equation of type: $f(u)+f^{-1}(u)=u-7$ Can I use this in any way?

Or if we take $f(x)=x+2^x+\log_2{x}\space\space\space\space\space \forall x\in(0,\infty).$$f(x)$ is strictly increasing on $(0,\infty).$ So we have an unique solution for $f(x)=7$ but how could I find it? or atleast approximate it?

Answer 1

The function $g(u)=u + 2^u + \log_2{u}$ is strictly increasing, so it cuts every line $y=a$ at most once. I gues it is not difficult to guess for which $u$ if $a=7$.

Answer 2

Since $f$ increases, we see that $2$ is an unique root of $f(u)=7$.

Thus, $$x^2+3x-2=2$$ and we got the answer: $$\{1,-4\}$$