The quantum-mechanical hypervirial (HV) theorems for atomic bound and scattering states are expressed as the expectation values of the commutator [W, H] between a virial operator W and the system Hamiltonian H. They provide various relationships between the average kinetic and potential energies, as well as the scattering phase shifts. Since the virial integral with an approximate wavefunction generally contains the information on the first-order error of the solution, we examine the applicability of the HV theorems not only for the purpose of testing approximate solutions of the Schrödinger equations, but also improving the solution by optimizing the parameters in the trial functions. To illustrate the approach, extensive numerical studies have been carried out for typical two-electron atomic systems in their bound and scattering states. The study is especially warranted for the scattering states, where simple criteria to test the quality of solutions are not readily available. We show that, with judicious choices of W, the hypervirial tests can provide useful checks on the accuracy of approximate solutions, and a way to determine the optimal values for the (nonlinear) parameters in the approximate solutions. Several points of caution in applications of the HV tests are explained.