We prove that, for a free noncyclic group F, the second homology group H₂(FQ;Q/ is an uncountable Q-vector space, where FQ denotes the Q-completion of F. This solves a problem of AK Bousfield for the case of rational coefficients. As a direct consequence of this result, it follows that a wedge of two or more circles is Q-bad in the sense of Bousfield-Kan. The same methods as used in the proof of the above result serve to show that H₂ (FZ, Z) is not a divisible group, where FZ is the integral pronilpotent completion of F.