What is the Numerically Stable Way to Compute the Coefficients of a Quadratic Function from Three Points

Question

I would like to compute the coefficients a0, a1, and a2 of a quadradic function (polynomial of degree 2) given three points (x0, y0), (x1, y1), and (x2, y2), where yi = a0 + a1*xi + a2*xi*xi?

I have tried the following two formulas, but I am not really impressed with the precision of the output

final double x0mx1, x0mx2, x1mx2, t0, t1, t2;
double a2, a1, a0;

x0mx1 = (x0 - x1);
x0mx2 = (x0 - x2);
x1mx2 = (x1 - x2);


// method one
t0 = (y0 / (x0mx1 * x0mx2));
t1 = (y1 / (-x0mx1 * x1mx2));
t2 = (y2 / (x0mx2 * x1mx2));

a2 = (t0 + t1 + t2);
a1 = -((t0 * (x1 + x2)) + (t1 * (x0 + x2)) + (t2 * (x0 + x1)));
a0 = (t0 * x1 * x2) + (t1 * x0 * x2) + (t2 * x0 * x1);

// method two
a2 = ((((y1 - y0) * (x0mx2)) + ((y2 - y0) * ((-x0mx1)))) /
                (((x0mx2) * ((x1 * x1) - (x0 * x0)))
                + (((-x0mx1)) * ((x2 * x2) - (x0 * x0)))));
a1 = (((y1 - y0) - (a2 * ((x1 * x1) - (x0 * x0)))) / ((-x0mx1)));
a0 = y0 - (a2 * x0 * x0) - (a1 * x0);

The results do sort of fit, i.e., seem roughly to be within a +/- 1e-5 * max{ |a0'|, |a1'|, |a2'| } window of the real solution a0', a1', and a2'.

Is there a better, more numerically stable way to compute the coefficients?

I am using Java, btw, although I think this does not matter.

Cheers, Thomas.