2007 AMC 12A Problem 21

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)

For

ax^2+bx+c,

the sum of zeros is

-\tfrac ba

and the product is

\tfrac ca

Setting these equal gives

c=-b,

so the sum of coefficients

a+b+c

simplifies

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.

Problem 21 in other years