Problem:
The quadratic equation x2+mx+n=0 has roots that are twice those of x2+px+m=0, and none of m,n, and p is zero. What is the value of n/p?
Answer Choices:
A. 1
B. 2
C. 4
D. 8
E. 16
Solution:
Let r1​ and r2​ be the roots of x2+px+m=0. Since the roots of x2+mx+n= 0 are 2r1​ and 2r2​, we have the following relationships:
m=r1​r2​,n=4r1​r2​,p=−(r1​+r2​), and m=−2(r1​+r2​)
So
n=4m,p=21​m, and pn​=21​m4m​=(D)8​.
OR
The roots of
(2x​)2+p(2x​)+m=0
are twice those of x2+px+m=0. Since the first equation is equivalent to x2+2px+4m=0, we have
m=2p and n=4m, so pn​=8
Answer: D​.
The problems on this page are the property of the MAA's American Mathematics Competitions