0NATURE OF ROOTS
In quadratic equation ax2 + bx + c = 0, the term b2 – 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 D1 and D2 be the discriminants of two quadratic equations, then
(i) If D1 + D2 ³ 0, then (a) At least one of D1 and D2 ³ 0. (b) If D1 < 0 then D2 > 0
(ii) If D1 + D2 < 0, then (a) At least one of D1 and D2 < 0. (b) If D1 > 0 then D2 < 0.
Brain Demur…
F The quadratic trinomial ax2 + bx + c = 0 will be a perfect square if D = 0 i.e. b2 – 4ac = 0
F If a is a repeated root of the ax2 + 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 ax2 + 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 ax2 + bx + c = 0 are rational numbers, then these roots must be integers.
