9 points on a quadric hypersurface

The number of degree monimals in $P^3$ are $x_ix_j$ where $0\le i \le j\le 3$ so there are $10$ of them. Each quadric is linear combination of them so it is parametrized by 10 variables. Now suppose you are given 9 points. To find the suitable quadric, you need to solve the coefficients and you get a linear system of equations. Since the number of variables is more than the number equations, you always get a nonzero solution.