In this work, we propose a Bayesian source separation method of linear-quadratic (LQ) and linear mixtures. Since our method relies on truncated prior distributions, it is particularly useful when the bounds of the sources and of the mixing coefficients are known in advance; this is the case, for instance, in non-negative matrix factorization. To implement our idea, we consider a Gibbs' sampler equipped with latent variables, which are set to simplify the sampling steps. Experiments with synthetic data point out that the new proposal performs well in situations where classical ICA-based solutions fail to separate the sources. Moreover, in order to illustrate the application of our method to actual data, we consider the problem of separating scanned images.