We present a new iterative procedure for solving nonlinear equations with multiple roots with high efficiency. Starting from the arithmetic mean of Newton’s and Chebysev’s methods, we generate a two-step scheme using weight functions, resulting in a family of iterative methods that satisfies the Kung and Traub conjecture, yielding an optimal family for different choices of weight function. We have performed an in-depth analysis of the stability of the family members, in order to select those members with the highest stability for application in solving mathematical chemistry problems. We show the good characteristics of the selected methods by applying them on four relevant chemical problems.