Nature of the roots of the Quadratic Equation ax² + bx + c, a ≠ 0, a, b, c ∈ R

To find out the nature of the roots of a quadratic equation, we find the value of the discriminant D \[D=b^2-4ac\]

1. If D < 0, we immediately say that equation has no solution in R.


2. If D > 0, then we examine whether it is a perfect square of a rational number or not.
   1. If D > 0 and D is not a perfect square, then roots are real and distinct.
   2. If D > 0 and it is a perfect square of a rational number and a, b, c ∈ Q, then the roots are real and rational. Roots are also distinct.


3. If D = 0 then the roots are real and equal and if a, b, c ∈ Q the roots are equal rational numbers.