Given two vector bundles E and F on a variety X and a morphism from Sym^2(E) to F, we compute the cohomology class of the locus in X where the kernel of this morphism contains a quadric of prescribed rank. This formula has many applications to moduli theory of which we mention: (i) a simple proof of Borcherds' and Pandharipande's result that the Hodge class on the moduli space of polarized K3 surfaces of fixed genus is of Noether-Lefschetz type, (ii) an explicit canonical divisor on the Hurwitz space parametrizing degree k covers of the projective line from curves of genus 2k-1, (iii) myriads of effective divisors of small slope on the moduli space of curves. This is joint work with Rimanyi.