2005 AMC 12B Problem 12

Below is the professionally curated solution for Problem 12 of the 2005 AMC 12B, from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 2005 AMC 12B solutions, or check the answer key.

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Concepts:Vieta’s Formulas quadratic

Difficulty rating: 1530

12.

The quadratic equation $x^2 + mx + n = 0$ has roots that are twice those of $x^2 + px + m = 0,$ and none of $m,$ $n,$ and $p$ is zero. What is the value of $\dfrac{n}{p}?$

$1$

$2$

$4$

$8$

$16$

Solution:

Let $r_1$ and $r_2$ be the roots of $x^2 + px + m = 0,$ so $m = r_1 r_2$ and $p = -(r_1 + r_2).$

The roots of $x^2 + mx + n = 0$ are $2r_1$ and $2r_2,$ so $n = 4r_1 r_2$ and $m = -2(r_1 + r_2).$

Then $n = 4m$ and $m = 2p,$ which gives $p = \dfrac{m}{2},$ so $\dfrac{n}{p} = \dfrac{4m}{\tfrac{m}{2}} = 8.$

Thus, the correct answer is D.

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