We consider an optical and mechanical mode interacting through both linear and quadratic dispersive couplings in a general cavity-optomechanical set-up. The parity and strength of an intrinsic quadratic optomechanical coupling (QOC) provides an opportunity to control the optomechanical (OM) interaction. We quantify this interaction by studying normal-mode splitting (NMS) as a function of the QOC's strength. The proposed scheme exhibits NMS features equivalent to a hybrid-OM system containing either an optical parametric amplifier or a Kerr medium. Such a system in reality could offer an alternative platform for devising state-of-art quantum devices with requiring no extra degrees-of-freedom as in hybrid-OM systems.