# Location of Roots

### Location of Roots

1. If both roots of equation ax² + bx + c = 0 represent opposite signs then

(i) D = b² – 4ac > 0

(ii) Product of roots of equation is less than zero

2. If both of equation ax² + bx + c = 0 have same sign then two possibilities

(i) If both roots of equation ax² + bx + c = 0 are positive then

(a) D = b² – 4ac ≥ 0

(b) Product of roots of equation is greater than zero i,e  α .β = c/a > 0

(c) Sum of roots is greater than zero i.e α + β = – b/a > 0

(ii) If both roots of equation ax² + bx + c = 0 are negative then

(a) D = b² – 4ac ≥ 0

(b) Product of roots of equation is greater than zero i.e α.β = c/a > 0

(c) Sum of roots is less than zero i.e α + β = – b/a < 0

Example: Value of m for which both root of equation x² – mx + 1 = 0 are less than unity.

Solution: Given,

x² – mx + 1 = 0

its corresponding expression is f(x) = x² – mx + 1 … (1)

ax² + bx + c = 0 … (2)

compare equation (1) and (2)

a = 1, b = – m, c = 1

given equation have roots less then unity, then

i) D ≥ 0

D = b² – 4ac > 0

m² – 4 (1)(1) ≥ 0

m² – 4 ≥ 0

m² ≥ 4

m ≥ ± 2

m ϵ (-∞, -2] È [ 2, ∞) … (1)

ii) f(x) = x² – mx + 1

Put x = 1

f(1) > 0

(1-m+1) > 0

(2-m) > 0

m – 2 < 0

m < 2

m ϵ (-∞, 2] … (ii)

iii) x² – mx + 1 = 0

f(x) = x² – mx + 1 = 0

ax² + bx + c = 0

Compare above equations

a = 1, b = – m, c = 1

-b /2a < 1

m/2 < 1

m < 2

m ϵ (-∞, 2]

from (i) (ii) and (iii) m ϵ (-∞, 2].