Most binary forms come from a pencil of quadrics

A pair of symmetric bilinear forms A and B determine a binary form f(x,y) := \operatorname {disc}(Ax-By). We prove that the question of whether a given binary form can be written in this way as a discriminant form generically satisfies a local-global principle and deduce from this that most binary forms over \mathbb{Q} are discriminant forms. This is related to the arithmetic of the hyperelliptic curve z^2 = f(x,y). Analogous results for nonhyperelliptic curves are also given.