Finding Roots of Bivariate Polynomial Surfaces: A Slice Technique Approach

[MostlyConfusd]

Is there a formula for finding the roots of a bivariate polynomial in x and y with the form:

(a^2)xy+abx+acy+bc

Where a, b, and c are constants, of course.

 

[chiro]

Hey MostlyConfusd and welcome to the forum.

You are trying to find the roots for a multi-variable surface (which is a simple one in terms of a surface), so what I might suggest is you analyze through slicing.

When you pick a particular slice, you will have a uni-variate polynomial equation. The key however to root solving (for real roots to occur) is that the discriminant must be positive or zero.

Consider now using the slice technique where you set up a polynomial (where you choose either your x or y as your slice) and consider the regions where the determinant is positive and this will tell you where the roots have to exist.

From there you can decide whether you get a point or a line for the solutions. A point will imply that only one slice gives a positive discriminant but a line implies you get many slices with positive discriminants.