The Standard Model of particle physics describes weak interactions mediated by massive gauge bosons that interact with each other in well-defined ways. Observations of the production and decay of WW, WZ, and ZZ boson pairs are an opportunity to check that these self-interactions agree with the Standard Model predictions. Furthermore, final states that include quarks are very similar to the most prominent final state of Higgs bosons produced in association with a W or Z boson. Diboson production where WW is a significant component has been observed at the Tevatron collider in semi-hadronic decay modes. We present a search for ZW and ZZ production in a final state containing two charged leptons and two jets using 8.9 fb−1 of data recorded with the CDF detector at the Tevatron. We select events by identifying those that contain two charged leptons, two hadronic jets, and low transverse missing energy ( E /T ). We increase our acceptance by using a wide suite of high- pT lepton triggers and by relaxing many lepton identification requirements. We develop a new method for calculating corrections to jet energies based on whether the originating parton was a quark or gluon to improve the agreement between data and the Monte Carlo simulations used to model our diboson signal and dominant backgrounds. We also make use of neural-network-based discriminants that are trained to pick out jets originating from b quarks and light-flavor quarks, thereby increasing our sensitivity to Z → bb ̄ and W/Z → qq ̄' decays, respectively. The number of signal events is extracted through a simultaneous fit to the dijet mass spectrum in three channels: a heavy-flavor tagged channel, a light-flavor tagged channel, and an untagged channel. We measure σZW/ZZ = 2.5+2.0-1.0 pb, which is consistent with the SM cross section of 5.1 pb. We establish an upper limit on the cross section of σZW/ZZ