2005 AMC 12B Problem 12

Problem 12 of 25IntermediateAlgebra

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}?

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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.

More practice

Concepts: Vieta’s Formulas · quadratic

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