Suppose we have two quadratic equations in $\mathbb R[x_1,\dots, x_n]$. What is the expected dimension of their intersection?

In general what can we say about intersection of $k$ quadratics? How many intersections we expect no real solutions in the best case (which I think should be treated similar to $\mathbb C$ setting)? Is there a way to understand with Bezout`s theorem?

This is what I have. Given a set of solutions ($0$ dimensional points planted) I can generate many equations of degree $2$ that are 'random' looking which have these as common solutions. How many should I generate in any reasonable sense to have only these 0 dimensional points as solutions? I was thinking $O(1)$ should suffice for most purposes (which is best case for me). However may be we need as many as $Ω(n)$ where $n$ is number of variables (which will make my system infeasible)? Is there rigorous analysis we can make?