Let $\emptyset\neq X\subset\mathbb{P}^{n}$ be an algebraic variety such that $$ X=V(F_{1},\ldots,F_{m}) $$ for certain linearly independent homogeneous polynomials $F_{1},\ldots,F_{m}\in K[X_{0},\ldots,X_{n}]$ of degree $2$.
Let us suppose that $G_{1},\ldots,G_{m}\in K[X_{0},\ldots,X_{n}]$ are homogeneous polynomials of the same degree $d$ such that $$ X= V(G_{1},\ldots,G_{m}). $$ I would like to prove that $d=2$. Is it true? How could I prove it? In case it is not true, are there easy conditions we should impose for the result to hold?