  In this paper we consider the problem of bounding the Betti numbers,
$b_i(S)$, of a semi-algebraic set $S \subset \R^k$ defined by polynomial
inequalities $P_1 \geq 0,...,P_s \geq 0$, where $P_i \in \R[X_1,...,X_k]$ and
$\deg(P_i) \leq 2$, for $1 \leq i \leq s$. We prove that for $0\le i\le k-1$,
\[ b_i(S) \le{1/2}(\sum_{j=0}^{min\{s,k-i\}}{{s}\choose j}{{k+1}\choose
{j}}2^{j}). \] In particular, for $2\le s\le \frac{k}{2}$, we have \[ b_i(S)\le
{1/2} 3^{s}{{k+1}\choose {s}} \leq {1/2} (\frac{3e(k+1)}{s})^s. \] This
improves the bound of $k^{O(s)}$ proved by Barvinok. This improvement is made
possible by a new approach, whereby we first bound the Betti numbers of
non-singular complete intersections of complex projective varieties defined by
generic quadratic forms, and use this bound to obtain bounds in the real
semi-algebraic case.
