  In this paper we establish three results on small-height zeros of quadratic
polynomials over $\overline{\mathbb Q}$. For a single quadratic form in $N \geq
2$ variables on a subspace of $\overline{\mathbb Q}^N$, we prove an upper bound
on the height of a smallest nontrivial zero outside of an algebraic set under
the assumption that such a zero exists. For a system of $k$ quadratic forms on
an $L$-dimensional subspace of $\overline{\mathbb Q}^N$, $N \geq L \geq
\frac{k(k+1)}{2}+1$, we prove existence of a nontrivial simultaneous
small-height zero. For a system of one or two inhomogeneous quadratic and $m$
linear polynomials in $N \geq m+4$ variables, we obtain upper bounds on the
height of a smallest simultaneous zero, if such a zero exists. Our
investigation extends previous results on small zeros of quadratic forms,
including Cassels' theorem and its various generalizations and contributes to
the literature of so-called "absolute" Diophantine results with respect to
height. All bounds on height are explicit.
