Inspired by inertial methods and extragradient algorithms, two algorithms were proposed to investigate fixed point problem of quasinonexpansive mapping and pseudomonotone equilibrium problem in this study. In order to enhance the speed of the convergence and reduce computational cost, the algorithms used a new step size and a cutting hyperplane. The first algorithm was proved to be weak convergence, while the second algorithm used a modified version of Halpern iteration to obtain strong convergence. Finally, numerical experiments on several specific problems and comparisons with other algorithms verified the superiority of the proposed algorithms.