Cardinality of the zero locus of a degree 2 homogeneous polynomial on Z/2Z: avoiding Chevalley-Warning

Question

I have never developed sufficient knowledge in algebraic geometry but I ran into an apparently easy problem, so I apologise in advance if my question sounds naive. Suppose to have a degree $2$ homogeneous (non-trivial) polynomial $p(\mathbf{x})=\sum\limits_{j=1}^n\sum\limits_{i=1}^ja_{ij}x_ix_j\in\mathbb{F}_2[\mathbf{x}]$ of $n$ variables, with $\mathbb{F}_2=\mathbb{Z}/2\mathbb{Z}=\{0,1\}$. If $n\geq3$, we can use Chevalley-Warning theorem to conclude that the zero locus of $p$ must have even cardinality, whence odd cardinality of the zero locus of $p$ may occur only in the cases $n=1,2$ (for $n=1$, $p(x)=x^2$ does the job while for $n=2$ we have four possibilities, $p(x)=xy,x(x+y),(x+y)y,x^2+xy+y^2$). Do you think that Chevalley-Warning theorem can be avoided in this particular case, to prove that for $n\geq3$ the zero locus has even cardinality?