Complex Numbers And Quadratic Equations question 758
Question: If $ P(x)=ax^{2}+bx+c $ and $ Q(x)=-ax^{2}+dx+c $ where $ ac\ne 0 $ , then $ P(x).Q(x)=0 $ has at least [IIT 1985; Pb. CET 2003; AMU 2005]
Options:
A) Four real roots
B) Two real roots
C) Four imaginary roots
D) None of these
Show Answer
Answer:
Correct Answer: B
Solution:
Let all four roots are imaginary. Then roots of both equations $ P(x)=0 $ and $ Q(x)=0 $ are imaginary. Thus $ b^{2}-4ac<0;d^{2}+4ac<0 $ , So $ b^{2}+d^{2}<0 $ , which is impossible unless $ b=0,d=0 $ . So, if $ b\ne 0 $ or $ d\ne 0 $ at least two roots must be real. If $ b=0, $ $ d=0 $ , we have the equations. $ P(x)=ax^{2}+c=0 $ and $ Q(x)=-ax^{2}+c=0 $ or $ x^{2}=-\frac{c}{a};x^{2}=\frac{c}{a} $ as one of $ \frac{c}{a} $ and $ -\frac{c}{a} $ must be positive, so two roots must be real.
BETA
