AMC 2007 12A P21

Problem Statement

The sum of the zeros, the product of the zeros, and the sum of the coefficients of the function $f(x)=ax^{2}+bx+c$ are equal. Their common value must also be which of the following?

$\textrm{(A)} \textrm{the coefficient of }x^{2}$

$\textrm{(B)} \textrm{the coefficient of }x$

$\textrm{(C)} \textrm{the y-intercept of the graph of }y=f(x)$

$\textrm{(D)} \textrm{one of the x-intercepts of the graph of }y=f(x)$

$\textrm{(E)} \textrm{the mean of the x-intercepts of the graph of }y=f(x)$

Problem Link

Solution

$$f(x)=ax^{2}+bx+c = a(x-r_1)(x-r_2) = ax^2 - a(r_1+r_2) + ar_1r_2$$ $$a + b + c = r_1r_2 = r_1 + r_2$$ $$a + b + c = a - a(r_1 + r_2) + ar_1r_2$$ $$a + b + c = a(1 - (r_1 + r_2) + (r_1r_2)) = a(1 - (a+b+c) + (a+b+c))$$ $$\therefore a + b + c = a$$