A useful improvement upon currently available procedures is presented for solving the type of equation indicated in the title. Such equations arise, for instance, in minimum reflux calculations for multicomponent distillation problems. Those engaged in improving the performance of distillation software will find the most use for the proposed method. It requires no explicit step size limits, is more robust than Newton's method, cannot converge to a root other than the one being sought, and retains the quadratic convergence exhibited by Newton's method.