Let's suppose for a real variable $x$ we find the equation $x = (f(x) + a)(f(x)+b)$ for some invertible function $f(x).$
Can we find all solutions to this through partial fixed points or partial inverses, similar to how you might solve $0 = (x+a)(x+b)?$
For instance, we solve for $x = f(x)+a$ and independently also solve $x = f(x) + b$ and thus find all solutions?
My guess is that the answer is no, but I'm hoping there are analytical shortcuts that might approximate the solutions for a given $f(x)$, like convenient Taylor or integral approximations?
Hint
Only to think better, take a particular case $a=b=0$. In this case you have
$$x=[f(x)]^2.$$
Your numerical solution for this equation clearly depends on $f$.
For instance, if $f(x)$ is a constant functions and equal to $c$, then you find
$$x=c^2.$$
For each value you take for $c$, you find a different solution for your equation.