This work presents two new numerical methods for finding the zeros of nonlinear equations, based on the Adomian decomposition approach and the quadrature rule. Using a predictor–corrector scheme, the proposed iterative methods provide a high order of convergence and do not require second derivatives, thereby reducing computational time. The methods are tested on several functions, and their efficiency is demonstrated through comparisons with various established techniques. To support the numerical findings, graphical analyses are provided. Furthermore, a study of the basins of attraction is conducted to validate the stability of the proposed methods.