Complex Numbers And Quadratic Equations question 458
Question: If $ b_1b_2 $ =2( $ c_1+c_2 $ ), then at least one of the equations $ x^{2}+b_1x+c_1=0 $ and $ x^{2}+b_2x+C_2=0 $ has
Options:
A) imaginary
B) real roots
C) purely imaginary roots
D) none of these
Show Answer
Answer:
Correct Answer: B
Solution:
[b] $ LetD_1andD_2 $ be desicriminants of $ x^{2}+b_1x+c_1=0 $ $ andx^{2}+b_2x+c_2=0,respectively.Then, $ $ D_1+D_2=b_1^{2}-4c_1+b_2^{2}-4c_2 $ $ =(b_1^{2}+b_2^{2})-4(c_1+c_2) $ $ =b_1^{2}+b_2^{2}-2b_1b_2[\therefore b_1b_2=2(c_1+c_2)] $ $ ={{(b_1-b_2)}^{2}}\ge 0 $
$ \Rightarrow D_1\ge 0orD_2\ge 0orD_1andD_2 $ both are positive Hence, at least one of the equations has real roots.
BETA
