We use three-terminal magnetic tunnel junctions (MTJs) designed for field-free switching by spin-orbit torques (SOTs) to systematically study the impact of dual voltage pulses on the switching performance. We show that the concurrent action of an SOT pulse and an MTJ bias pulse allows for reducing the critical switching energy below the level typical of spin-transfer-torque while preserving the ability to switch the MTJ on the subnanosecond time scale. By performing dc and real-time electrical measurements, we discriminate and quantify three effects arising from the MTJ bias: the voltage-controlled change of the perpendicular magnetic anisotropy, current-induced heating, and the spin-transfer torque. The experimental results are supported by micromagnetic modeling. We observe that, depending on the pulse duration and the MTJ diameter, different effects take a lead in assisting the SOTs in the magnetization-reversal process. Finally, we present a compact model that allows for evaluating the impact of each effect due to the MTJ bias on the critical switching parameters. Our results provide input to optimize the switching of three-terminal devices as a function of time, size, and material parameters