  In this paper, we first prove that a quadratic form from $\mathbb{S}^m$ to
$\mathbb{S}^n$ is non-harmonic biharmonic if and only if it has constant energy
density $(m+1)/2$. Then, we give a positive answer to an open problem
concerning the structure of non-harmonic biharmonic quadratic forms. As a
direct application, using classification results for harmonic quadratic forms,
we infer classification results for non-harmonic biharmonic quadratic forms.
