We study an optomechanical (OM) system with optical and mechanical modes interacting through a dispersive linear and quadratic optomechanical coupling (QOC) in an unresolved sideband limit. The presence of a QOC plays a crucial role in altering the effective OM interaction and the spring constant of the mechanical resonator. The choice of a positive QOC in the system increases the spring constant, inducing a 'stiffer mode.' The existence of this stiffer mode leads to unconventional behavior in the total displacement noise consisting of imprecision, backaction, and thermal noises. We further show that unlike the conventional OM system wherein the optimal total displacement noise is limited by the imprecision and backaction noises, it is due to the competing imprecision and thermal noise, making it feasible to observe displacement sensing below the conventional standard quantum limit.