Complex Numbers And Quadratic Equations question 29
Question: If a, b be the roots of the quadratic equation $ ax^{2}+bx+c=0 $ and $ k $ be a real number, then the condition so that $ \alpha <k<\beta $ is given by
Options:
A) $ ac>0 $
B) $ ak^{2}+bk+c=0 $
C) $ ac<0 $
D) $ a^{2}k^{2}+abk+ac<0 $
Show Answer
Answer:
Correct Answer: D
Solution:
Here $ ax^{2}+bx+c=a(x-\alpha )(x-\beta ) $ Since $ \alpha ,\beta $ be the roots of $ ax^{2}+bx+c=0 $ . Also $ \alpha <k<\beta , $ so $ a(k-\alpha )(k-\beta )<0 $ Also $ a^{2}k^{2}+abk+ac=a(ak^{2}+bk+c) $ $ =a^{2}(k-\alpha )(k-\beta )<0 $ Þ $ a^{2}k^{2}+abk+ac<0 $
BETA
