The purpose of this paper is to provide strong reformulations for binary quadratic problems. We propose a first methodological analysis on a family of reformulations combining Dantzig-Wolfe decomposition and Quadratic Convex Reformulation principles. As a representative case study, we apply them to a cardinality constrained quadratic knapsack problem, providing extensive experimental insights. We show that a few reformulations of our family yield to continuous relaxations that are strong in terms of dual bounds and computationally efficient to optimize. We report and analyze in depth a particular reformulation providing continuous relaxations whose solutions turn out to be integer optima in all our tests. We finally discuss on how to exploit these results in a generic binary quadratic programming context.