How to find a simple algorithm that takes 6 points for two curves (degree of 2 to 4) and determines the point of intersection

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I have 6 points each from two curves (equation and polynomial degrees not given) and I want to find an algorithm that gives me the intersection point of the two curves. Note that the given points are not integers but decimals to 4 significant places.

In my search to address this I came across Determine Polynomial equation from given set of data points by Finite Difference which is very informative and solves my problem partly using the following steps:

  1. checking point differences till constant difference is found to determine polynomial degree
  2. using coefficient of first term to get a certain number of equations with unknowns
  3. putting the found values back to arrive at equation

I am not sure how to go about the later part of finding the roots of a polynomial with decimal coefficients after I equate the 2 polynomials.

I am generally very open to a completely different way to solve my whole problem. Though the video solved the first part, I feel that there could be a simpler or more efficient way to go about it that may not require as much computation since I'm interested in an algorithm.

UPDATE

I know all this may sound like a lot but in a nutshell what I'm trying to achieve is, Finding the intersection of two curves given only 6 points each that lie on each curve.