We all know that the discriminant is the part $b^2-4ac$ of the equation
that we use to find the roots of a quadratic equation eg: $ax^2+bx+c=0$ or the part in a trinomial expression like $ax^2+bx+c$.
What I want to know is what kind of ideas we can conclude for a given inequality of a discriminant. We know that in quadratic formulas the number of real roots depends on the case whether the discriminant is <0, 0, or >0. But if given only the discriminant and the inequality of it, for examples like this:
$ b^2-4ac<0 $
What can we conclude by the given information for each case? How can it help to reveal certain problems involving trinomial expressions?
Thank you in advance. It will be really nice if you could solve my problem and provide links that can lead me to study further on this topic.