# 2007 AMC 12A Problems/Problem 21

## Problem

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?

$\textrm{(A)}\ \textrm{the\ coefficient\ of\ }x^{2}~~~ \textrm{(B)}\ \textrm{the\ coefficient\ of\ }x$ $\textrm{(C)}\ \textrm{the\ y-intercept\ of\ the\ graph\ of\ }y=f(x)$ $\textrm{(D)}\ \textrm{one\ of\ the\ x-intercepts\ of\ the\ graph\ of\ }y=f(x)$ $\textrm{(E)}\ \textrm{the\ mean\ of\ the\ x-intercepts\ of\ the\ graph\ of\ }y=f(x)$

## Solution

By Vieta's formulas, the sum of the roots of a quadratic equation is $\frac {-b}a$, the product of the zeros is $\frac ca$, and the sum of the coefficients is $a + b + c$. Setting equal the first two tells us that $\frac {-b}{a} = \frac ca \Rightarrow b = -c$. Thus, $a + b + c = a + b - b = a$, so the common value is also equal to the coefficient of $x^2 \Longrightarrow \fbox{\textrm{A}}$.

To disprove the others, note that:

• $\mathrm{B}$: then $b = \frac {-b}a$, which is not necessarily true.
• $\mathrm{C}$: the y-intercept is $c$, so $c = \frac ca$, not necessarily true.
• $\mathrm{D}$: an x-intercept of the graph is a root of the polynomial, but this excludes the other root.
• $\mathrm{E}$: the mean of the x-intercepts will be the sum of the roots of the quadratic divided by 2.