In a work of C. Gutierrez and V. Guiñez, local problems around a class of rank-2 singular points called simple such as normal forms, finite determinacy, and versal unfoldings are studied for smooth positive quadratic differential forms on surfaces, as well as for their associated pair of foliations (with singularities). To extend this study to the class of rank-2 singular points, two cases of rank-2 singular points remain to be treated, namely that of type C and that of type E($\lambda$), with $ \lambda \geq 1 $. Using the theory of normal forms for singularities of positive quadratic differential forms, we obtain the phase portrait and a versal unfolding for type C singular points proving that their codimension is three.