This a joint work with Wanke Yin. Let $M\subset \mathbb{C}^{n+1}$ ($n\ge 2$) be a real analytic submanifold defined by an equation of the form: $w=|z|^2+O(|z|^3)$, where we use $(z,w)\in {CC}^{n}\times CC$ for the coordinates of ${C}^{n+1}$. We first derive a pseudo-normal form for $M$ near $0$. We then use it to prove that $(M,0)$ is holomorphically equivalent to the quadric $(M_\infty: w=|z|^2,\ 0)$ if and only if it can be formally transformed to $(M_\infty,0)$, using the rapid convergence method. We also use it to give a necessary and sufficient condition when $(M,0)$ can be formally flattened. Our main theorem generalizes a classical result of Moser for the case of $n=1$.