Second-harmonic generation (SHG) is investigated for a dispersive medium with quadratic and cubic nonlinearities. All possible solutions assuming a zero initial value for the second harmonic are obtained and tabulated. Based on the analysis, the optimal condition for SHG is deduced and the dependence of the SHG conversion efficiency upon the nonlinearity, the dispersion, and the input field intensity is discussed. Also, a thorough numerical study of the behavior of SHG efficiency is performed and the results are reported. A technique for estimation of the unknown third-order nonlinearity coefficients of a material from SHG observation is proposed.