2000 AMC 10 Problem 24 — Solution | LIVE by Po-Shen Loh

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2000 AMC 10 Problem 24

Below is the professionally curated solution for Problem 24 of the 2000 AMC 10, from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 2000 AMC 10 solutions, or check the answer key.

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Concepts:function substitution Vieta’s Formulas

Difficulty rating: 1690

24.

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 $f(3z) = 7.$

$-\dfrac13$

$-\dfrac19$

$0$

$\dfrac59$

$\dfrac53$

Solution:

To evaluate $f(3z),$ set $\dfrac{x}{3} = 3z,$ so $x = 9z.$ Then $f(3z) = (9z)^2 + 9z + 1 = 81z^2 + 9z + 1.$

Setting this equal to $7$ gives $81z^2 + 9z - 6 = 0.$

By the sum-of-roots formula, the sum of the values of $z$ is $-\dfrac{9}{81} = -\dfrac19.$

Thus, the correct answer is B.

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