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].