We study the convergence to equilibrium of a class of nonlinear recombination models. In analogy with Boltzmann's H theorem from kinetic theory, and in contrast with previous analysis of these models, convergence is measured in terms of relative entropy. The problem is formulated within a general framework that we refer to as Reversible Quadratic Systems. Our main result is a tight quantitative estimate for the entropy production functional. Along the way we establish some new entropy inequalities generalizing Shearer's and related inequalities.