Question

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?

the coefficient of $x^2$

the coefficient of $x$

the $y$ -intercept of the graph of $y=f(x)$

one of the $x$ -intercepts of the graph of $y=f(x)$

the mean of the $x$ -intercepts of the graph of $y=f(x)$

Answer: A

Concepts:Vieta’s Formulas quadratic

Difficulty rating: 1660

Small Hint:

For $ax^2+bx+c,$ the sum of zeros is $-\tfrac ba$ and the product is $\tfrac ca$

Big Hint:

Setting these equal gives $c=-b,$ so the sum of coefficients $a+b+c$ simplifies

Solution:

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.

2007 AMC 12A Problem 21

Answer: A

Difficulty rating: 1660

Small Hint:

Big Hint:

Solution:

Problem 21 in Other Years