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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
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.
$\Rightarrow 0\leq (a-c)^2<(a+b+c)(a+c-b)$
$\Rightarrow a+c<b$, 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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