Interpretation of CP violation measurements using charmonium decays, in both the B̄0 and B̄s0 systems, can be subject to changes due to "penguin" type diagrams. These effects can be investigated using measurements of the Cabibbo-suppressed B̄0→J/ ψπ+π- decays. The final state composition of this channel is investigated using a 1.0 fb-1 sample of data produced in 7 TeV pp collisions at the LHC and collected by the LHCb experiment. A modified Dalitz-plot analysis is performed using both the invariant mass spectra and the decay angular distributions. An improved measurement of the B ̄0→J/ψπ+π- branching fraction of (3.97±0.09±0.11±0.16)×10-5 is reported where the first uncertainty is statistical, the second is systematic, and the third is due to the uncertainty of the branching fraction of the decay B-→J/ψK- used as a normalization channel. Significant production of f0(500) and ρ(770) resonances is found in the substructure of the J/ψπ+π- final state, and this indicates that they are viable final states for CP violation studies. In contrast evidence for the f0(980) resonance is not found. This allows us to establish the first upper limit on the branching fraction product B(B̄0→J/ψf0(980))×B(f 0(980)→π+π-)<1.1×10 -6, leading to an upper limit on the absolute value of the mixing angle of the f0(980) with the f0(500) of less than 31, both at 90% confidence level. © 2013 CERN, for the LHCb Collaboration. Published by the American Physical Society under the terms of the http://creativecommons.org/licenses/by/3.0/ Creative Commons Attribution 3.0 License. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.