This paper addresses the characterization of optimal input distributions for the general additive quadrature Gaussian-mixture (GM) noise channel under an average power constraint. The considered model can be used to represent a wide variety of communication channels, such as the well-known Bernoulli-Gaussian and Middleton Class-A impulsive noise channels, co-channel interference in cellular communications, and cognitive radio channels under imperfect spectrum sensing. We first demonstrate that there exists a unique input distribution achieving the channel capacity and the optimal input has an uniformly distributed phase. By using the Kuhn-Tucker conditions (KTC) and Bernstein's theorem, we then demonstrate that there are always a finite number of mass points on any bounded interval in the optimal amplitude distribution. Equivalently, the optimal amplitude input distribution is discrete. Furthermore, by applying a novel bounding technique on the KTC, it is then shown that the optimal amplitude distribution has a finite number of mass points.