We propose a hybrid algorithm that combines discrete ellipsoid-based search (DEBS) and a branch-and-cut (B&C) MIP solver to solve binary quadratic programming (BQP) problems, an important class of optimization problems with a number of practical applications. We perform experiments on benchmark instances for the BQP problem and compare the performance of two B&C based solvers, the DEBS method that is commonly used in the communications community, and the new hybrid algorithm. Our experimental results demonstrate that the new hybrid algorithm outperforms both the well-known MIP solvers and the DEBS approach. Further comparison against two state-of-the-art special-purpose algorithms in the literature demonstrates that the hybrid approach is competitive: achieving the same or better performance on six of seven benchmark sets against one algorithm and performing competitively against the semi-definite programming (SDP) based algorithm for moderate size problems and some dense problems, while under-performing on larger problems.