This paper presents DuoBFT, a Byzantine fault-tolerant protocol that uses trusted components to provide commit decisions in the Hybrid fault model in addition to commit decisions in the BFT model. By doing so, it enables the clients to choose the response fault model for its commands. Internally, DuoBFT commits each client command under both the hybrid and Byzantine models, but since hybrid commits take fewer communication steps and use smaller quorums than BFT commits, clients can benefit from the low-latency commits in the hybrid model. DuoBFT uses a common view-change change protocol to handle both fault models. To achieve this, we enable a notion called Flexible Quorums in the hybrid fault model by revisiting the quorum intersection requirements in hybrid protocols. The flexible quorum technique enables having a hybrid view change quorum that is of the same size as a BFT view-change quorum. This paves a path for efficiently combining both the fault models within a single unified protocol. Our evaluation on a wide-area deployment reveal that DuoBFT can provide hybrid commits with 30% lower latency to existing protocols without sacrificing throughput. In absolute terms, DuoBFT provides sub-200-millisecond latency in a geographically replicated deployment.