I'm looking for solutions to the equation
$$x^2+2^x+x^x = 12$$
Which is satisfied obviously by $x=2$ and somewhat less obviously by $x\approx-3.4512$. By plotting $|z^2 + 2^z + z^z - 12|$ on the complex field these solutions are visible, along with potential complex solutions.
My questions are
- What operations could be applied to this equation to transform it into a solvable form? This looks impossible with only elementary functions but surely there exist some artificially constructed functions that are relatively well-known/accepted by the Maths community that are capable of reducing this to a solvable form. Each component ($x^x$, $2^x$ and $x^2$) is solvable on its own but the sum renders the inverse functions (Lambert W and square root operator) useless.
- Why is it that the negative solution, a seemingly transcendental number, is perfectly real and also bears no relation to the other root, $2$.
Website used: https://jutanium.github.io/ComplexNumberGrapher/
I did almost the same as you did but I considered instead the contour levels of $$\Phi(a,b)= \bigg[\Re \Big[ f(a+i b)\Big]\bigg]^2+\bigg[\Im \Big[ f(a+i b)\Big]\bigg]^2$$which are interesting.
The only way I found to solve the problem is the minimization of $$\Phi(a,0)=\Im\left(a^a+a^2+2^a\right)^2+\left(\Re\left(a^a+a^2+2^a\right)-12 \right)^2$$
The problem is that the best point is $$a_*=-3.4511944449422438390521195349718938696027536302639\cdots$$ but $$\Phi(a_*,0)=0.000189008\cdots$$
Computing $$a_*^2+2^{a_*}+a_*^{a_*} - 12=0.0000481154+0.0137479456 \,i$$
Back to the initial contour plot, do it for $-3.452 \leq a \leq -3.450$ and $-0.001 \leq b \leq 0.001$ and ask for the level $0.0001$ or lower to observe the discontinuity.
This looks to be a false root.