Intelligent reflecting surface (IRS), a recently introduced technology, enhances spectral and energy efficiency by intelligently adjusting the propagation conditions between a base station (BS) and mobile users (MUs). An IRS consists of many low-cost passive reflecting elements, each of which can be controlled to change the phase of the incident signal to improve the quality of the received signal. In this paper, we study the problem of power control at the BS for the IRS-aided physical-layer broadcasting under the quality of service (QoS) constraints. Our goal is to minimize the transmit power at the BS by jointly designing the transmit beamforming at the BS and the phase shifts of the passive elements at the IRS, subject to each MUs signal-to-noise ratio constraint which characterizes MU's QoS. Furthermore, to validate the proposed optimization methods, we derive lower bounds of the minimum transmit power at the BS with respect to the number of MUs, the number of IRS elements, and the number of antennas at the BS. The simulation results demonstrate that the transmit power at the BS in the IRS-aided system is very close to the lower bound, and is significantly lower than the transmit power of conventional schemes without IRS.