please help me to solve the following problem.
Let $F$ be a non-abelian free group and $w(x)=1$ be an equation in one variable $x$ ($w(x)$ may contain elements of $F$ as constants). Clearly, one can consider $w(x)$ as an element of free product $F\ast \langle x\rangle$.
Suppose $w(a)=1$ for all $a\in F$.
Is it true that $w(x)$ equals $1$ in $F\ast \langle x\rangle$?
Do you know a simple proof? Probably, you can remember the papers which can be useful for this propblem?