 # What are we learning ?

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 .

# First look at the knowledge graph. # 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
6/10
##### Suggested Book
Challenges and Thrills in Pre College Mathematics Excursion Of Mathematics

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, 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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