  In this paper, we consider the characterization of the bentness of quadratic
Boolean functions of the form $f(x)=\sum_{i=1}^{\frac{m}{2}-1}
Tr^n_1(c_ix^{1+2^{ei}})+ Tr_1^{n/2}(c_{m/2}x^{1+2^{n/2}}) ,$ where $n=me$, $m$
is even and $c_i\in GF(2^e)$. For a general $m$, it is difficult to determine
the bentness of these functions. We present the bentness of quadratic Boolean
function for two cases: $m=2^vp^r$ and $m=2^vpq$, where $p$ and $q$ are two
distinct primes. Further, we give the enumeration of quadratic bent functions
for the case $m=2^vpq$.
