Newton method is a well-known tool for solving finite-dimensional systems of equations. Pure Newton-Raphson method has at least quadratic convergence rate, but its convergence radius is often limited. Damped version of Newton method uses smaller step-size with same direction, with larger convergence ball but linear convergence rate. We propose mixed step-size choice strategy, incorporating both quadratic convergence rate and wide (global in some cases) convergence radius. The method can be used in cases of under-determined equations and Banach-space equations. We present a modification of proposed method requiring no a-priori knowledge of problem constants as well. The method may be also used for solving a class of non-convex optimization problems.