In this article, the author present a complete classification for the parametric surfaces of bidegree (1, 2) over the real field. We also provide some results for the inverse problem: given a segment of a line or twisted cubic curve, research look for a patch (1,2) which includes this segment as a subset of its singular locus. For instance we characterize the ruled surfaces containing a twisted cubic curve such that all generating lines cut twice the cubic curve, which are indeed parametric surfaces of bidegree (1,2).