Singular quadrics, covariants of binary quartics and $\operatorname{SL}_2(\Bbbk)$ representation

Question

Consider the space of irreducible $\operatorname{SL}_2(\Bbbk)$ representation: $$R_4 = \langle x^4, x^3y, x^2y^2,xy^3, y^4 \rangle.$$ This is a 5 dimensional vector space, and we have decomposition for quadrics in this space: $$S^2(R_4) = R_8\oplus R_4 \oplus R_0.$$ There are some singular quadrics. How one can define by the element $(f_8,f_4,f_0)$ that corresponding quadri is singular. I think this could be connected with covariants of binary forms, but I don't know where to start. Thanks for any ideas.