Finding quadratic function from two points and known max value

[Dave Britten]

I need to work this into a C# program for generating points along an arc, and the math is making me cross-eyed.

Essentially, I know two points on the parabola, I know the y value of the maximum (but not x), and I know that the second derivative is negative, i.e. the coefficient of x^2 is negative. All the typical approaches of treating it as a system of 3 linear equations seem to rely on having three known points, but I've got two, plus an additional equation 2*a*x_3+b=0 where x_3 is unknown.

Is there a straight-forward way of tackling this without pulling a bunch of numeric and matrix libraries into the program?