2009 AMC 12A Problem 9

Problem 9 of 25EasierAlgebra

Suppose that

f(x + 3) = 3x^2 + 7x + 4

and

f(x) = ax^2 + bx + c.

What is

a + b + c?

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Solution

Note that

a + b + c = f(1).

Using

f(x + 3) = 3x^2 + 7x + 4

with

x = -2

gives

\begin{aligned} f(1) &= f(-2 + 3) \\ &= 3(-2)^2 + 7(-2) + 4 \\ &= 12 - 14 + 4 = 2. \end{aligned}

Thus, the correct answer is D.

Concepts: function · substitution

Problem text and solution from the LIVE past-contest archive. See also the AoPS wiki page for community solutions.