**NATURE OF ROOTS **

In quadratic equation ax^{2} + bx + c = 0, the term b^{2} – 4ac is called discriminant (D or *D)* .

**(1)** **If a, b, c **

**Î**

*R*and*a***¹**

**0, then :**

(i) If *D* > 0, then roots are real and distinct (ii) If *D* = 0, then roots are real and equal

(iii) If *D* ³ 0, then roots are real . (iv) If *D* < 0, then roots are complex .

**(2)** **If a, b, c **

**Î**

*Q*,*a***¹**

**0, then :**

(i) If *D* > 0 and *D* is a perfect square then roots are rational.

(ii) If *D* > 0 and *D* is not a perfect square then roots are irrational .

**(3)** **Conjugate roots: **The irrational and complex roots of a quadratic equation always occur in pairs. Therefore

(i) If one root be a + ib then other root will be a - ib.

(ii) If one root be a + Öb then other root will be a - Öb.

**(4)** **If D_{1} and D_{2} be the discriminants of two quadratic equations, then**

(i) If D_{1} + D_{2} ³ 0, then (a) At least one of D_{1} and D_{2} ³ 0. (b) If D_{1} < 0 then D_{2} > 0

(ii) If D_{1} + D_{2} < 0, then (a) At least one of D_{1} and D_{2} < 0. (b) If D_{1} > 0 then D_{2} < 0.

*Brain Demur…*

F *The quadratic trinomial ax ^{2} + bx + c = 0 will be a perfect square if D = 0 i.e. b^{2} – 4ac = 0*

F *If **a** is a repeated root of the ax ^{2} + bx + c = 0 then *

*a*

*is a root of equation f’(x) = 0 as well.*

F *If a, b, c* ÏR then* the roots need not be conjugate.*

F *If a, b, c are irrational *then* the roots need not be conjugate.*

F *If a + b + c = 0 then one root is always unity and the other root is , c/a*

**ROOTS UNDER PARTICULAR CONDITIONS **

For the quadratic equation ax^{2} + bx + c = 0.

(1) b = 0 Þ roots are of equal magnitude but of opposite sign.

(2) c = 0 Þ one root is zero, other is –b/a.

(3) b = c = 0 Þ both roots are zero.

(4) a = c Þ roots are reciprocal to each other.

(5) a + b + c = 0 Þ one root is 1 and second root is c/a.

(6) a = b = c = 0, then equation will become an identity and will be satisfied by every value of x.

(7) a = 1 and b, c Î I and the root of equation ax^{2} + bx + c = 0 are rational numbers, then these roots must be integers.