An effective scheme within two displaced bosonic operators with equal positive and negative displacements is extended to study qubit-oscillator systems analytically in a unified way. Many previous analytical treatments, such as generalized rotating-wave approximation （GRWA） [Phys. Rev. Lett. 99, 173601 （2007）] and an expansion in the qubit tunneling matrix element in the deep strong coupling regime [Phys. Rev. Lett. 105, 263603 （2010）] can be recovered straightforwardly within the present scheme. Moreover, further improving GRWA and the extension to the finite-bias case are implemented easily. The algebraic formulae for the eigensolutions are then derived explicitly and uniquely, which work well in a wide range of the coupling strengths, detunings, and static bias including the recent experimentally accessible parameters. The dynamics of the qubit for an oscillator in the ground state is also studied. At the experimentally accessible coupling regime, GRWA can always work well. When the coupling is enhanced to the intermediate regime, only the improving GRWA can give the correct description, while the result of GRWA shows strong deviations. The previous Van Vleck perturbation theory is not valid to describe the dynamics in the present-day experimentally accessible regime, except for the strongly biased cases.
