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 $\displaystyle f(x)=ax^{2}+bx+c$ are equal. Their common value must also be which of the following?

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 $x^2 \Longrightarrow \textrm{A}$ .

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.