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Competency in Focus: Discriminant of the quadratic equation and basic inequalities  This problem from I.S.I. B.Stat. Entrance 2017 is a problem on quadratic equation and some basic inequalities .

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Next understand the problem

Let a , b , c be real numbers such that a+b+c<0. Suppose that equation \( ax^2+bx+c=0 \) has imaginary roots .Then ,
Source of the problem
I.S.I. B.Stat. Entrance 2016, UGA Problem 26
Key Competency
Discriminant of the quadratic equation and basic inequalities 
Difficulty Level
6/10
Suggested Book
Challenges and Thrills in Pre College Mathematics Excursion Of Mathematics 

Start with hints

Do you really need a hint? Try it first!
Given that the quadratic equation \( ax^2+bx+c=0\) has imaginary roots then what can we say about it’s discriminant ?
Discriminant of the quadratic equation having imaginary roots is less than zero . So, we have $D<0\Rightarrow 0\leq b^2 <4ac$
So $a,c$ have same sign.
Also,$b^2<(a+c)^2-(a-c)^2$
$\Rightarrow 0\leq (a-c)^2<(a+b+c)(a+c-b)$
$\Rightarrow a+c<b$, as $a+b+c<0$. Now we will check the options.
This shows, if $a>0,c>0$ then $b>0$, contradicting $a+b+c<0$. Hence $a<0,c<0$.

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