$f\in k[x_1,\ldots,x_n]$ degree $2$ homogeneous polynomial having a non-trivial zero in odd deg extension of $k$ , $f$ has a non-trivial zero in $k$.

Question

The following is an exercise in Lang's Algebra, Chapter 5, #28.

Let $f$ be a homogeneous degree 2 polynomial in $n$ variables over a field $k$. If $f$ has a non-trivial zero in an extension of odd degree then $f$ has a non-trivial zero in $k$.

Ewan Delanoy gave an answer here for $n=2$ case. I just want to see this result for $n=3$, using the $n=2$ case. For this consider $f(x,y,z)\in k[x,y,z]$ homogeneous of degree $2$, and without loss of generality $f(x,y,1)\in k[x,y]$ has a nontrivial zero in $L^2$, where $\mid L:k\mid$ is odd. Cannot apply induction ($n=2$ case) immediately since $f(x,y,1)$ may not be homogeneous. How can I settle this? Thanks.