  The intersection cut paradigm is a powerful framework that facilitates the
generation of valid linear inequalities, or cutting planes, for a potentially
complex set S. The key ingredients in this construction are a simplicial conic
relaxation of S and an S-free set: a convex zone whose interior does not
intersect S. Ideally, such S-free set would be maximal inclusion-wise, as it
would generate a deeper cutting plane. However, maximality can be a challenging
goal in general. In this work, we show how to construct maximal S-free sets
when S is defined as a general quadratic inequality. Our maximal S-free sets
are such that efficient separation of a vertex in LP-based approaches to
quadratically constrained problems is guaranteed. To the best of our knowledge,
this work is the first to provide maximal quadratic-free sets.
