If I am working with a polynomial and guess a root $x_0$, I can divide the polynomial by $x-x_0$ to obtain a simpler polynomial without that one root.
Can I do something similar for a system of equations? Suppose I have $n$ quadratic equations in $n$ variables. I know, say $2n$ solutions but I suspect there are more. Can I somehow "eliminate" the known solutions simplifying the system?