f[x_]:=a*x^2+b*x+c
Factor[Solve[{f[x[A]]==y[A],f[x[B]]==y[B],f[x[C]]==y[C]},{a,b,c}]]

{{a->((x[B] y[A]-x[C] y[A]-x[A] y[B]+x[C] y[B]+x[A] y[C]-x[B] y[C])/((-x[A]+x[B]) (x[B]-x[C]) (-x[A]+x[C]))),b->((-((x[B])^2) y[A]+((x[C])^2) y[A]+((x[A])^2) y[B]-((x[C])^2) y[B]-((x[A])^2) y[C]+((x[B])^2) y[C])/((x[A]-x[B]) (x[A]-x[C]) (x[B]-x[C]))),c->(((x[B])^2) x[C] y[A]-x[B] ((x[C])^2) y[A]-((x[A])^2) x[C] y[B]+x[A] ((x[C])^2) y[B]+((x[A])^2) x[B] y[C]-x[A] ((x[B])^2) y[C])/((x[A]-x[B]) (x[A]-x[C]) (x[B]-x[C]))}}