2007 AMC 12A Problem 21

The product of the zeros is $\tfrac ca$ and the sum of the zeros is $-\tfrac ba.$ Equating them gives $c=-b.$ Then the sum of the coefficients is $a+b+c=a,$ which is the coefficient of $x^2.$ The other choices fail in general: for $f(x)=-2x^2-4x+4$ the common value is $-2,$ but the coefficient of $x$ is $-4,$ the $y$-intercept is $4,$ the $x$-intercepts are $-1\pm\sqrt3,$ and their mean is $-1.$ Thus, the correct answer is A.