  Equations in free groups have become prominent recently in connection with
the solution to the well known Tarski Conjecture. Results of Makanin and
Rasborov show that solvability of systems of equations is decidable and there
is a method for writing down in principle all solutions. However, no practical
method is known; the best estimate for the complexity of the decision procedure
is P-space.
  The special case of one variable equations in free groups has been open for a
number of years, although it is known that the solution sets admit simple
descriptions. We use cancellation arguments to give a short and direct proof of
this result and also to give a practical polynomial time algorithm for finding
solution sets. One variable equations are the only general subclass of
equations in free groups for which such results are known.
  We improve on previous attempts to use cancellation arguments by employing a
new method of reduction derived from techniques from formal language theory.
Our paper is self contained; we assume only knowedge of basic facts about free
groups.
