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?