Exact expressions for the magnetization or the dipole moment, the quadrupolar moment and the Curie temperature of the spin-3/2 Blume-Capel model on the Bethe lattice are derived using the recursion method. The thermal variations of the magnetization and quadrupolar moment order parameters are studied in depth for a range of the coupling parameter, alpha = D/J, and a number of characteristic behaviors for their thermal variations are obtained. Besides the stable branches of the order parameters, we find the unstable branches of them which occur at low temperatures and near the first- and second-order phase-transition temperatures. Finally, the phase diagram and the critical behavior of the model are discussed. (C) 2002 Elsevier Science B.V. All rights reserved.