We show that for any finite-rank–free group (Formula presented.), any word-equation in one variable of length (Formula presented.) with constants in (Formula presented.) fails to be satisfied by some element of (Formula presented.) of word-length (Formula presented.). By a result of the first author, this logarithmic bound cannot be improved upon for any finitely generated group (Formula presented.). Beyond free groups, our method (and the logarithmic bound) applies to a class of groups including (Formula presented.) for all (Formula presented.), and the fundamental groups of all closed hyperbolic surfaces and 3-manifolds. Finally, using a construction of Nekrashevych, we exhibit a finitely generated group (Formula presented.) and a sequence of word-equations with constants in (Formula presented.) for which every nonsolution in (Formula presented.) is of word-length strictly greater than logarithmic.