How can we find the coefficients of a polynomial given the coordinates of 3 points?

Question

I tried solving the system of $3$ equation for this, and it's actually very hard. I'm just trying to find the coefficients $A,B,C$, of the polynomial $Ax^2+Bx+C$ given the points $(a,b) (c,d) (e,f)$.

Answer 1

Suppose the three points are $(x_1, y_1), (x_2, y_2) , (x_3, y_3) $

Then it follows that

$ A x_1^2 + B x_1 + C = y_1 $

$ A x_2^2 + B x_2 + C = y_2 $

$ A x_3^2 + B x_3 + C = y_3 $

Using Cramer's rule, the three parameters $A,B,C$ are given by

$ A = \dfrac{1}{D} \begin{vmatrix} y_1 && x_1 && 1 \\ y_2 && x_2 && 1 \\ y_3 && x_3 && 1 \end{vmatrix} $

$ B = \dfrac{1}{D} \begin{vmatrix} x_1^2 && y_1 && 1 \\ x_2^2 && y_2 && 1 \\ x_3^2 && y_3 && 1 \end{vmatrix} $

$ C = \dfrac{1}{D} \begin{vmatrix} x_1^2 && x_1 && y_1 \\ x_2^2 && x_2 && y_2 \\ x_3^2 && x_3 && y_3 \end{vmatrix} $

where

$D = \begin{vmatrix} x_1^2 && x_1 && 1 \\ x_2^2 && x_2 && 1 \\ x_3^2 && x_3 && 1 \end{vmatrix} $