2000 AMC 10 Problem 24

Problem 24 of 25HarderAlgebra

Let

f

be a function for which

f\left(\dfrac{x}{3}\right) = x^2 + x + 1.

Find the sum of all values of

z

for which

f(3z) = 7.

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Solution

To evaluate

f(3z),

set

\dfrac{x}{3} = 3z,

x = 9z.

Then

\begin{aligned} f(3z) &= (9z)^2 + 9z + 1 \\ &= 81z^2 + 9z + 1. \end{aligned}

Setting this equal to

7

gives

81z^2 + 9z - 6 = 0.

By the sum-of-roots formula, the sum of the values of

z

-\dfrac{9}{81} = -\dfrac19.

Thus, the correct answer is B.