I read the following statement in Conway's book on quadratic forms and tried to prove it:
A function $f$ is a quadratic form if and only if $f(a\textbf {v}) = a^2 f(\textbf{v}) $ and $B (\textbf{v},\textbf{w}) := f(\textbf{v} + \textbf{w}) - f(\textbf{v}) - f(\textbf{w})$ is a bilinear form.
(Here $\textbf {v}$ is a real number $n$ dimensional vector. The function $f$ is a quadratic form if $f( (x_1,x_2, \cdots, x_n)) = \sum_{i=1}^{n} \sum_{j=1}^n \alpha_{ij} x_i x_j$ for constants $\alpha_{ij}$)
I proved necessity, but I am having problems with sufficiency. I know the function will be of the form $f( (0,\cdots,x_i,\cdots,0)) = \alpha_{ii} (x_i)^2$, for vectors with only one non-zero coordinate, but I had trouble proving that the $\alpha_{ij}$ are constants for $i \neq j$.
If $B(v,w)$ is a bilinear form, then $B(v,v)$ is a quadratic form. Observe that if $B(v,w)=f(v+w)-f(v)-f(w)$ then $B(v,v)=f(2v)-2f(v)=4f(v)-2f(v)=2f(v).$ Therefore $f(v)$ is a quadratic form.