1. MATHEMATICAS
i)Relation between roots and the coefficients.
ii)Nature of the roots of quadratic equations.
Presentation by
D.Rohini
I B.Sc. Mathematics(R)
2. RELATION BETWEEN THE ROOTS AND COEFFICIENTS
Let 𝜶 𝟏, 𝜶 𝟐, … 𝜶 𝒏 be the roots of the equation
𝒂 𝟎 𝒙 𝒏 + 𝒂 𝟏 𝒙 𝒏−𝟏 + 𝒂 𝟐 𝒙 𝒏−𝟐 + ⋯ 𝒂 𝒏
𝑺 𝟏 = 𝒊=𝟏
𝒏
𝜶𝒊 = (−𝟏)
𝒂 𝟏
𝒂 𝟎
𝑺 𝟐 = 𝒊,𝒋
𝒏
𝜶𝒊 𝜶𝒋 = (−𝟏) 𝟐 𝒂 𝟐
𝒂 𝟎
𝑺 𝒏 = 𝜶 𝟏, 𝜶 𝟐, … , 𝜶 𝒏 = (−𝟏) 𝒏 𝒂 𝒏
𝒂 𝟎
where 𝑺 𝒏denotes the sum of the product of n roots at a time.
3. NATURE OF THE ROOTS OF A QUADRATIC EQUATION
The quadratic equation : 𝒂𝒙 𝟐
+ 𝒃𝒙 + 𝒄 = 𝟎
The roots are given by
𝒙 =
−𝒃± 𝒃 𝟐−𝟒𝒂𝒄
𝟐𝒂
4. NATUREOFTHEROOTSOFAQUADRATIC
EQUATION
Let ∆ = 𝒃 𝟐
− 𝟒𝒂𝒄
Case i)if ∆ = 𝑏2 − 4𝑎𝑐 = 0,then the roots are real and
equal.
Case ii)if ∆ = 𝑏2 − 4𝑎𝑐 > 0,then the roots are real and
distinct.
Case iii)if ∆ = 𝑏2 − 4𝑎𝑐 <0,then the roots are
imaginary.
5. EXAMPLES
Case I : 𝒙 𝟐 + 𝟒𝒙 + 𝟒 = 𝟎
here ∆ = 𝑏2 − 4𝑎𝑐 = 0,therefore the roots are real and
equal.
Case II : 𝒙 𝟐 + 𝟓𝒙 + 𝟔 = 𝟎
here ∆ = 𝑏2
− 4𝑎𝑐 > 0,therefore the roots are real and
distinct.
Case III : 𝒙 𝟐 + 𝒙 + 𝟏 = 𝟎
here ∆ = 𝑏2 − 4𝑎𝑐 <0,therefore the roots are
imaginary.
