In this work, we consider the problem of blind source separation (BSS) by departing from the usual linear model and focusing on the linear-quadratic (LQ) model. We propose two provably robust and computationally tractable algorithms to tackle this problem under separability assumptions which require the sources to appear as samples in the data set. The first algorithm generalizes the successive nonnegative projection algorithm (SNPA), designed for linear BSS, and is referred to as SNPALQ. By explicitly modeling the product terms inherent to the LQ model along the iterations of the SNPA scheme, the nonlinear contributions of the mixing are mitigated, thus improving the separation quality. SNPALQ is shown to be able to recover the ground truth factors that generated the data, even in the presence of noise. The second algorithm is a brute-force (BF) algorithm, which is used as a post-processing step for SNPALQ. It enables to discard the spurious (mixed) samples extracted by SNPALQ, thus broadening its applicability. The BF is in turn shown to be robust to noise under easier-to-check and milder conditions than SNPALQ. We show that SNPALQ with and without the BF postprocessing is relevant in realistic numerical experiments.