Question on quadratic forms I know a theorem which says: 

If a non-singular quadratic form (homogeneous polynomials of degree $2$) over a field $K$ represents zero non-trivially (i.e., there is a nontrivial solution of the equation $F=0$), then it represents all the non-zero elements of field $K$. 

A quadratic form $F$ is said to represent $a$ (in $K$) if there exists $x_1,x_2,\dots,x_n$ such that $F(x_1,x_2,\dots,x_n)= a$.  

My question is whether the converse of the theorem is true when char $K=0$.

When $K$ is a finite field of order $3$, then $X^2+Y^2$ contradicts the converse of the theorem. 

You can take $x^2$ over $\mathbb{C}$.
