We study second order ordinary differential equations of Newton type with integrals of motion that depend quadratically on the velocity. In particular, we introduce the class of cofactor pair systems, which admit two quadratic integrals of motion of a special form. It is shown that this implies that the system in fact admits a full set of Poisson commuting integrals of motion, and consequently is completely integrable. Methods are given for testing whether a given Newton system belongs to this class, and for constructing infinite families of cofactor pair systems. Several known result about separable potentials are included in the theory as special cases. As an application, it is shown how to extend the classical concept of Stäckel separability to a class of time-dependent potentials.