1. Math 20-1 Chapter 4 Quadratic Equations
Teacher Notes
3.1 Factoring Quadratic Equations

2. x2 + (a + b)x + ab = (x + a)(x + b)
Solve by Factoring Simple Trinomials
Simple Trinomials: The coefficient of the x2 term is 1.
Recall: (x + 6)(x + 4) = x(x + 4) + 6(x + 4)
= x2 + 4x + 6x + (6)(4)
= x2 + (4 + 6)x + (6)(4)
= x2 + 10x + 24
The last term is the
product of the
constant terms.
Explain each step going backwards
The middle term is the sum of
the constant terms of the binomial.
x2 + (a + b)x + ab = (x + a)(x + b)
Factor:
x2 + 9x + 20
Factors of 20 1 20 2 10 4 5
x2 + (5 + 4)x + (4)(5)
x2 + 5x + 4x + (4)(5)
x(x + 5) + 4(x + 5)
(x + 5)(x + 4)
4.2A.3

3. Factors of 12 1 12 2 6 3 4
4.2A.4

4. Factoring General Trinomials Decomposition
Recall: (3x + 2)(x + 5)
= 3x(x + 5) +2(x + 5)
= 3x2 + 15x + 2x + 10
= 3x2 + 17x + 10
Explain each step going backwards
To factor 3x2 + 17x + 10, find two numbers
that have a product of 30 and a sum of 17.
Factors of 30 1 30 2 15 3 10
4.2A.5

5. Solving General Trinomials - the Decomposition Method
Factors of 24 1 24 2 12 3 8 4 6
3x2 - 10x + 8
The product is 3 x 8 = 24.
The sum is -10.
3x2 - 6x - 4x + 8
Rewrite the middle term of the
polynomial using -6 and -4.
(-6x - 4x is just another way of
expressing -10x.)
3x(x - 2)
- 4(x - 2)
Factor by grouping.
(x - 2)(3x - 4)
4.2A.6

6. Factoring General Trinomials - Some Examples
1. 6x2 + x - 15
product is - 90
sum is 1
10 & -9
= 6x2 + 10x - 9x - 15
= 2x(3x + 5)
- 3(3x + 5)
Check using foil: (3x + 5)(2x - 3)
= 6x2 - 9x + 10x + 15
= 6x2 + x - 15
= (3x + 5)(2x - 3)
y2 - 5y - 3
product is -36
sum is -5
4 & -9
= 12y2 - 9y + 4y - 3
= 3y(4y - 3)
+ 1(4y - 3)
= (4y - 3)(3y + 1)
3. 3x2 + 20x + 12
product is 36
sum is 20
2 & 18
= 3x2 + 18x + 2x + 12
= 3x(x + 6)
+ 2(x + 6)
= (x + 6)(3x + 2)

7. Factoring General Trinomials - the Inspection Method
1. List the pairs of factors
of the first and last terms.
Factor: x2 - 11x + 3
2 x 1= 2
3 x 3 = 9
Sum = 11
1 x 1 = 1
6 x 3 = 18
Sum = 19
1 x 3 = 3
6 x 1 = 6
Sum = 9
2. Do a cross product sum to
find the sum equal to the
middle term. 1 6 2 3 1 3
3. Place the terms in the
brackets, keeping in mind
FOIL. (place in opposite
brackets.)
Not equal to the middle
term so don’t use.
Try again.
This sum is equal to the
middle term, so use these
numbers.
Not equal to the middle
term so don’t use.
Try again.
(2x - 3)(3x - 1)
Check using foil:
6x2 - 2x - 9x + 3
= 6x2 - 11x + 3

8. Factor
Factors of 4 1 4 2
4.2A.7

9. Recall that, when multiplying conjugate binomials, the
Difference of Squares
Recall that, when multiplying conjugate binomials, the
product is a difference of squares.
(x - 7)(x + 7) = x2 + 7x – 7x - 49
= x2 - 49
Therefore, when factoring a difference of squares, the
factors will be conjugate binomials.
Factor: x
= (x - 9)(x + 9)
16x
= (4x - 11)(4x + 11)
(x)2 - (9)2
(4x)2 - (11)2
5x2 - 80y2
= 5(x2 - 16y2)
= 5(x - 4y)(x + 4y)
4.2A.8

10. Factoring a Difference of Squares with a Complex Base
Factor completely:
(x + y)
= B
= [B - 4]
[B + 4]
= [(x + y) - 4]
[(x + y) + 4]
= (x + y - 4)(x + y + 4)
25 - (x + 3)2
= [5 - (x + 3)]
[5 + (x + 3)]
= (5 - x - 3)(5 + x + 3)
= (-x + 2)(8 + x)
4.2A.9

11. 25(x - 1)2 - 4(3x + 2)2
= [5(x - 1) - 2(3x + 2)] [5(x - 1) + 2(3x + 2)]
= (5x x - 4)(5x x + 4)
= (-x - 9)(11x - 1)
Assignment
Suggested Questions:
Part A p
PartB (7-10)ac,11a,12a 13b 14 15a 16b 18c 19 mc
4.2A.10

12. Dealing with Fractions in Factoring

13. Greatest Common Factor Simple Trinomial Perfect Square Trinomial
4.2A Factoring Quadratic Expressions
Factor
Greatest Common Factor
Simple Trinomial
Perfect Square Trinomial
Difference of Squares
Decomposition
Fractions
Complex Bases
4.2A.2