In this paper we construct a variety - called complete - which allows us to study the enumerative problem to determine the numbers of quadrics given by certain conditions of simple contact to an i-space as an intersection problem on it. We start with these conditions and then we show how the degenerations related to them lead to a definition of the variety. Furtheron we determine the orbits of the PGL action on it regarding the ranks of matrices. Knowing the orbits we are able to show that the variety is complete with respect to the conditions mentioned above. The basic ideas of this paper were taken from the first named author's doctoral thesis (Halle 1968). Independently of our considerations I. Vainsencher offered a different approach to the problem, which is a generalization of the classical problem of complete conics: [\textit{I. Vainsencher}, Enumerative geometry and classical algebraic geometry, Prog. Math. 24, 199-235 (1982; Zbl 0501.14032)].