This paper introduces a novel application of frequency and amplitude formulation to determine the frequency of nonlinear Duffing oscillations. The formulation represents one of the simplest methods for estimating the relationship between frequency and amplitude. There is a pressing need for the swift calculation of the periodic properties of nonlinear oscillators in the applied sciences. Additionally, the paper addresses the challenge of identifying bifurcated solutions of the Duffing equation in space. A critical parameter value induces a bifurcation from a trivial solution to a nontrivial solution. The solutions obtained through the frequency-amplitude formulation are compared with results obtained through other perturbation methods documented in the literature. The proposed method has demonstrated its efficacy as a robust mathematical tool for various nonlinear oscillators, eliminating the necessity for linearization or perturbation. Furthermore, this method can be readily extended to address other nonlinear oscillating systems and nonlinear boundary value problems exhibiting bifurcations.