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We analytically and numerically investigate the dynamic propagation of the quadratically chirped Airy beams in right-handed material (RHM) and left-handed material (LHM) slabs. Based on the analytical expressions and numerical calculations, we find that through choosing the appropriate value of the quadratic chirp, one can modify the diffraction, the intensity distribution, the propagation path, and acceleration and deceleration of the quadratically chirped Airy beams to meet the actual usage accordingly. In addition, we also demonstrate the effect of the quadratic chirp on the Poynting vector and the angular momentum of the quadratically chirped Airy beams, respectively. Furthermore, we explore the gradient and scattering forces on a Rayleigh dielectric particles induced by the quadratically chirped Airy beams.