Page 11 - CHAPTER 4 (Quadratic equations)
P. 11
CHAPTER 4
QUADRATIC EQUATIONS
Case-II
When D = 0, Roots are real and equal and each root α= -b = β
2a
Case-III
When D < 0,
No real roots exist. Both the roots are imaginary.
Remark:
If D < 0, the roots are of the from a±ib (a,b∈R & i=√-1). If one root is a+ib,
then the other root will be a–ib,
e.g., x²-3x+12=0 has D = –39 < 0
∴ It’s roots are, α= -b+√D and β= -b-√D
2a 2a
3+√-39 3-√-39
α= and β=
2 2
3 i√39 3 i√39
α= + and β= = -
2 2 2 2
Example 9:
Find the nature of the roots of the following equations. If the real roots
exist, find them.
(i) 2x²-6x+3=0
(ii) 2x²–3x+5=0
Solution:
(i). The given equation 2x²-6x+3=0
Comparing it with ax²+bx+c=0, we get
a = 2, b = –6 and c = 3
∴ Discriminant, D = b² – 4ac = (–6)² – 4.2.3 = 36 – 24 = 12 > 0
∴ D > 0, roots are real and unequal.
11
