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Nature of Roots ( Quadratic Equation Pulkit
24/05/2018 0 0

NATURE 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…

The quadratic trinomial ax2 + bx + c = 0 will be a perfect square if D = 0 i.e. b2 – 4ac = 0

If a is a repeated root of the ax2 + bx + c = 0 then a is a root of equation f’(x) = 0 as well.

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

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

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.

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