We compute the minimal energy cost forextracting entanglement from the groundstate of a bosonic or fermionic quadraticsystem. Specifically, we find the minimalenergy increase in the system resulting from replacing an entangled pair of modes, sharing entanglement entropy S, by a product state, and we show how to construct modes achieving this minimal energy cost. Thus, we obtain a protocol independent lower bound on the extraction of pure state entanglement from quadratic systems. Due to their generality, our results apply to a large range of physical systems, as we discuss with examples