Reading Quadratic functions in geometry, topology, and M-theory by M. J. Hopkins and I. Singer (https://projecteuclid.org/download/pdf_1/euclid.jdg/1143642908), I reckon some unclarity on p. 410:
Suppose that $V$ is a vector space over $\mathbb{Q}$ of finite dimension, and $q_V:V→\mathbb{Q}$ is a quadratic function (not necessarily even) whose underlying bilinear form $B(x, y)=q_V(x+y)−q_V(x)−q_V(y)$ is non-degenerate. One easily checks that $q_V(x)−q_V(−x)$ is linear, and so there exists a unique $λ∈V$ with $q_V(x)−q_V(−x)=−B(λ, x)$.
EDIT 1) $q_V(x)−q_V(−x)$ is linear, as pointed out below. EDIT
2) How to achieve such a $\lambda$ (possibly without using linearity), maybe by exploiting non-degeneracy of $B$?