Complex Numbers And Quadratic Equations question 784
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) Real roots
B) Purely imaginary roots
C) Imaginary roots
D) None of these
Show Answer
Answer:
Correct Answer: A
Solution:
Let $ D_1 $ and $ D_2 $ be discriminants of $ x^{2}+b_1x+c_1=0 $ and $ x^{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,(\because b_1b_2=2(c_1+c_2)) $ = $ {{(b_1-b_2)}^{2}}\ge 0 $
Þ $ D_1\ge 0 $ or $ D_2\ge 0 $ or $ D_1 $ and $ D_2 $ both are positive.
BETA
