TITLE: Find the sum of all real solutions for $x$ to the equation $(x^2 + 2x + 3)^{(x^2+2x+3)^{(x^2+2x+3)}} = 2012.$
QUESTION [28 upvotes]: Find the sum of all real solutions for $x$ to the equation $(x^2 + 2x + 3)^{(x^2+2x+3)^{(x^2+2x+3)}} = 2012.$
I just know $x^{x^x}$ is increasing in $x$ and hence the equation has a unique solution, nut then I dont know how to move on, I also know viete' formula but I dont know if it helps here, thanks in advance.

REPLY [4 votes]: Sketch: $y^{y^y}=2012$ must have only one real solution, call it $a$ (we don't need to compute it). Then we are looking for the roots of $x^2 +2 x+3 - a = (x- x_0)(x-x_1)=x^2 - (x_0+x_1)x +x_0 x_1$ Hence...