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

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

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

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

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

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

Problem Link

Solution

$$f(x)=a{x}^2+bx+c=a(x-r_1)(x-r_2)=a{x}^2-a(r_1+r_2)+a{r}_1{r}_2$$ $$a+b+c=r_1{r}_2=r_1+r_2$$ $$a+b+c=a-a(r_1+r_2)+a{r}_1{r}_2$$ $$a+b+c=a(1-(r_1+r_2)+(r_1{r}_2))=a(1-(a+b+c)+(a+b+c))$$ $$\therefore a+b+c=a$$