Register for free to see more content

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 \(ax^2 + bx + c = 0\). It is denoted by the letter \(\Delta\) or \(D\).
Let us discuss the nature of the roots of the quadratic equation depending on the discriminant.
 
Case I: \(\Delta = b^2 - 4ac > 0\)
 
Here, \(b^2 - 4ac > 0\). That means the value of the discriminant is positive.
 
Then, the possible roots are b+b24ac2a and bb24ac2a.
If \(\Delta = b^2 - 4ac > 0\), then the roots are real and distinct.
 
Case II: \(\Delta = b^2 - 4ac = 0\)
 
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 \(\Delta = b^2 - 4ac = 0\), then the roots are real and equal.
 
Case III: \(\Delta = b^2 - 4ac < 0\)
 
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 \(\Delta = b^2 - 4ac < 0\), then there are no real roots.
Relation between roots and coefficients of a quadratic equation
If \(\alpha\) and \(\beta\) are the roots of the quadratic equation \(ax^2 + bx + c = 0\), then:
 
(i) Sum of the roots \(=\) α+β=ba
 
(ii) Product of the roots \(=\) αβ=ca
Quadratic equation \(=\) \(x^2 - (\text{sum of the roots})x + \text{product of the roots}\)
Important!
To know more about the relationship between roots and coefficients of a equation click here.