Roots and Coefficient of a Quadratic Equation

1. LESSON #7: Roots and Coefficient
of Quadratic Equation

2. Sum and Product of the Roots of a Quadratic
Equation
If r1 and r2 are the roots of the equation ax2 + bx + c = 0,
then
Sum of the roots: r1 + r2 = -
Product of the roots: r1 x r2 =

3. • USAGE:
Since, r₁ + r₂= - and r₁ r₂ = , therefore
the sum and product of roots are used in
checking
Based on the given
equation:
 x2 - 3x – 28 = 0
 a = 1; b = -3; c = -28
-> (x - 7)(x + 4) = 0
-> x – 7 = 0; x + 4 = 0
-> x = 7; x = -4
 Roots: 7 and -4
 r₁ + r₂ = -
r₁ + r₂ = 7+ (-4) = 3
- = - = 3
r₁ r₂ =
r₁ ∙ r₂ = 7 ∙ -4 = -28
= = -28

4. • USAGE:
Since, r₁ + r₂= - and r₁ r₂ = therefore it
has a relationship in writing quadratic
equations.
 Standard form:
 ax2 + bx + c = 0
 ax2 + bx + c = 0
a
x2 + x + = 0
x2-(- x) + = 0
x2 - (r₁ + r₂)x + (r₁ x r₂) = 0

5. EXAMPLE: Write the quadratic
equation given a pair of roots.
7, -2
x2 - (r₁ + r₂)x + (r₁ r₂) = 0
Where
x2 - (7 + -2)x + (7 -2) = 0
x2 - 5x -14 = 0

6. EXAMPLE: Write the quadratic
equation given a pair of roots.
5, 3
x2 - (r₁ + r₂)x + (r₁ r₂) = 0
Where
x2 - (5 + 3)x + (5 3) = 0
x2 - 8x + 15 = 0

7. EXAMPLE: Write the quadratic
equation given a pair of roots.
 - ,
x2 - (r₁ + r₂)x + (r₁ * r₂) = 0
Where
x2 - (- + )x + (- * ) = 0
x2 + (- ) = 0
x2 – 5 = 0

8. EXAMPLE: Write the quadratic
equation given a pair of roots.
 - ,
x2 - (r₁ + r₂)x + (r₁ * r₂) = 0
Where
x2 - (- + )x + (- * ) = 0
x2 + (- ) = 0
x2 – 3 = 0

9. Problem Analysis: Find the quadratic equation
whose roots are twice the roots of 6x2 - 5x + 1 = 0
The roots of the required quadratic equation are 2r₁ and 2r₂
of the original roots in the equation 6x2 - 5x +1 = 0.
 Therefore,
 x2 - (2r₁ + 2r₂)x + (2r₁ ∙ 2r₂) = 0
 x2 - 2(r₁ + r₂)x + 4(r₁ ∙ r₂) = 0
 x2 - 2(- )x + 4( ) = 0
 x2 - 2(- )x + 4( ) = 0
 x2 - x + = 0
 Or 3x2 - 5x + 2 = 0
x2 -(r₁ + r₂)x + (r₁ ∙ r₂)=0
a = 6; b = -5; c = 1

10. Assignment: Find the quadratic equation whose
roots are three more than the roots of
8x2 -18x – 9 = 0