# Nature of Roots of Quadratic Equation

## Nature of Roots of Quadratic Equation:

The expression, b2 – 4ac, connecting the coefficients of the quadratic equation ax2 + bx + c = 0, with a ≠ 0 determines the nature of the roots α and β of the equation and is known as the discriminant of the equation. It is generally denoted by D.

Several cases may arise as given below-

(i) If D > 0, but not a perfect square, then the roots are real, distinct and irrational.

(ii) If D = k2, k ∈ Q, k ≠ 0, then the roots are real, distinct and rational.

(iii) If D = 0, then the roots are real, rational and equal.

(iv) If D ≥ 0, then the roots are real.

(v) If D < 0, then the roots are imaginary.

(vi) If b = 0, then -b/a = 0 ⇒ α + β = 0, then the roots are equal in magnitudes but opposite in sign.

(vii) If c = a, then c/a = 1 ⇒ αβ = 1, then the roots are reciprocal to each other (provided b ≠ 0 in this case; since b = 0 implies ax2 + a = 0 ⇒ x2 = -1 ⇒ x = ±i).

(viii) If a + b + c = 0, then b = – (a + c)

The equation, then, reduces to ax2 – (a + c) x + c = 0

⇒ ax2 – ax – cx + c = 0 ⇒ ax (x – 1) – c (x – 1) = 0

⇒ (x – 1) (ax – c) = 0 ⇒ x = 1, c/a

Thus, if the sum of the coefficients of a quadratic equation is zero, then one of the roots is unity. In this case, D = b2 – 4ac = {-(a + c)}2 – 4ac = a2 + c2 – 2ac = (a – c)2, which is a perfect square and hence the roots are real distinct and rational.