Nature of Roots. A quadratic equation ax² + bx + c = 0, a ≠ 0, has two roots which by the quadratic formula are as under :

\(\frac{-b+\sqrt{b^2-4ac}}{2a}\) and \(\frac{-b-\sqrt{b^2-4ac}}{2a}\)

The expression b² – 4ac is called the discriminant. 

Examining the nature of the roots means to see what type of roots the equation has, that is, whether they are real or imaginary, real or irrational, equal or unequal. The nature of the roots depends entirely on the value of the discriminant D = b² – 4ac 

Thus, if a, b, c are rational, then

I. If D = b² – 4ac > 0 (i.e, positive), the roots are real and unequal. 

Also,

(a) If D = b² – 4ac is a perfect square, the roots are rational. 

(b) If D = b² – 4ac is not a perfect square, the roots are irrational. 

(c) If D = b² – 4ac = 0, the roots are equal, each being equal to \(\frac{-b}{2a}\). 

So, ax² + bx + c = 0 is a perfect square if D = 0. 

II. If D = b² – 4ac < 0 (i.e., negative), the roots are imaginary (complex).