We denote by QSL3 the family of quadratic differential systems possessing invariant straight lines, finite and infinite, of total multiplicity exactly three. In a sequence of papers the complete study of quadratic systems with invariant lines of total multiplicity at least four was achieved. In addition three more families of quadratic systems possessing invariant lines of total multiplicity at least three were also studied, among them the Lotka-Volterra family. However there were still systems in QSL3 missing from all these studies. The goals of this article are: to complete the study of the geometric configurations of invariant lines of QSL3 by studying all the remaining cases and to give the full classification of this family modulo their configurations of invariant lines together with their bifurcation diagram. The family QSL3 has a total of 81 distinct configurations of invariant lines. This classification is done in affine invariant terms and we also present the bifurcation diagram of these configurations in the 12-parameter space of coefficients of the systems. This diagram provides an algorithm for deciding for any given system whether it belongs to QSL3 and in case it does, by producing its configuration of invariant straight lines.