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=r1r2,n=4r1r2,p=−(r1+r2), and m=−2(r1+r2)
So
n=4m,p=21m, and pn=21m4m=8
\section*{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
The problems on this page are the property of the MAA's American Mathematics Competitions