Theory:

We have learnt that the roots of the quadratic equation \(ax^2 + bx + c = 0\) can be found by the quadratic formula:
 
x=b±b24ac2a
 
Important!
\(b^2 - 4ac\) is called the discriminant of the quadratic equation \(ax^2 + bx + c = 0\).
Let us discuss the nature of the roots of the quadratic equation depending on the discriminant.
 
Case I: \(b^2 - 4ac > 0\)
 
Here, \(b^2 - 4ac > 0\). That means the value of the discriminant is positive.
 
Then, the possible roots are b+b24ac2a and bb24ac2a.
If \(b^2 - 4ac > 0\), then the roots are real and distinct.
 
Case II: \(b^2 - 4ac = 0\)
 
Here, \(b^2 - 4ac = 0\). That means the value of the discriminant is zero.
 
x=b+02a and x=b02a
 
x=b2a and x=b2a
 
The possible roots are b2a and b2a.
If \(b^2 - 4ac = 0\), then the roots are real and equal.
 
Case III: \(b^2 - 4ac < 0\)
 
Here, \(b^2 - 4ac < 0\). That means the value of the discriminant is negative.
 
We won't get any real roots in this case.
If \(b^2 - 4ac < 0\), then there are no real roots.