I don't think your method works for finding the turning points. Finding where the force is zero is not the same as finding where the velocity is zero, which is where your turning points are. I would use turning point analysis to find the limiting values of x in the motion. That is, integrate the force to get the potential energy. Then set the potential energy equal to the total energy (define equilibrium to be zero potential energy; then the total energy is all kinetic at the equilibrium position) to find where the turning points are. Make sense?