Te Puna Ako Learning Centre
Factorising Quadratic Expressions of the form ax² + bx + c
To factorise means to write as factors. It is the opposite process to expanding.
Expanding:
(x + 2)(x + 3) = x² + 2x + 3x + 6 = x² + 5x + 6
Factorising
x² + 5x + 6 = (x + 2)(x + 3)
The numbers in the brackets multiply to give 6 (the constant in the quadratic expression)
and add to give 5 (the co-efficient of x in the quadratic expression).
Constant positive, co-efficient of x positive: x² + bx + c
When we are factorising x² + 7x + 12 we are looking for the pair of numbers that multiply to
give 12 and add to give 7 They are 3 and 4 so:
x² + 7x + 12 = (x + 3)(x + 4) or (x + 4)(x + 3) - (the factors can be in any order as
multiplication is commutative)
Exercise 1:
Factorise the following
1. x² + 6x + 8
2. x² + 8x + 7
3. x² + 5x + 4
4. a² + 12a + 36
5. a² + 13a + 36
6. a² + 4a + 4
7. b² + 14b + 49
8. b² + 11b + 18
9. b² + 9b + 18
10. x² + 8x + 15
Constant positive, co-efficient of x negative: x² - bx + c
In the example x² - 7x + 12 we are still looking for numbers that multiply to give 12 and add
to give 7 but because it is actually -7 the numbers must both be negative so:
x² - 7x + 12 = (x - 3)(x - 4)
Unitec: Document1
Exercise 2:
Factorise the following
1. x² - 9x + 14
2. r² - 6r + 9
3. r² - 11r + 24
4. t² - 9t + 20
5. a² -12a + 36
6. x² - 19x + 18
Constant negative, co-efficient of x negative or positive:
x² - bx - c and x² + bx -c
In the examples x² - x - 12 and x² + x - 12 we are looking for numbers that multiply to give
-12 so one must be negative and one most be positive. In the first example they must add
to give -1 and in the second example they must add to give 1. The easiest way to find the
number is:
Step 1:
Find Numbers
Find numbers that multiply to give 12 and subtract to give 1
Step 2
Decide whether positive or negative
If the coefficient of x is negative the bigger number will be negative. If the
co-efficient of x is positive the bigger number will be positive
x² - x – 12 = (x - 4)(x + 3) and x² + x - 12 = (x + 4)(x - 3)
Exercise 3
1. a²- a - 20
2. a² + a - 20
3. x² - 5x - 14
4. x² + 5x -14
5. x² - 8x - 20
6. a² + 2a – 48
Summary
If the constant is positive then the sign of numbers in the brackets will be the same and
the same as the sign of the co-efficient of x.
Look for numbers that multiply to give the constant and add to give the co-efficient of x.
Unitec: Document1
If the constant is negative then the sign of the numbers in the brackets will be different.
Look for numbers that multiply to give the constant and add or subtract to give the coefficient of x
If the co-efficient of x is negative then the biggest number will be negative. If the coefficient of x is positive then the biggest number will be positive.
Exercise 4
1. x² + 6x + 5
2. x² - 5x + 6
3. x² - 3x - 4
4. x² - x - 6
5. x² + 9x + 14
6. x² - 3x + 2
7. x² + x - 2
8. x² - x - 20
9. x² - 4x - 12
10. x² - 4x + 3
Answers
Exercise 1
1. (x + 4)(x + 2)
5. (a + 4)(a + 9)
9. (b + 6)(b + 3)
2. (x + 7)(x + 1)
6. (a + 2)(a + 2)
10. (x + 5)(x + 3)
3. (x + 4)(x + 1)
7. (b + 7)(b + 7)
4. (a + 6)(a + 6)
8. (b + 9)(b + 2)
2. (r - 3)(r - 3)
6. (x - 18)(x - 1)
3. (r - 8)(r - 3)
4. (t - 5)(t - 4)
2. (a + 5)(a - 4)
6. (a + 8)(a - 6)
3. (x - 7)(x + 2)
4. (x + 7)(x – 2)
2. (x - 2)(x - 3)
6. (x - 1)(x - 2)
10. (x - 1)(x - 3)
3. (x - 4)(x + 1)
7. (x + 2)(x - 1)
4. (x - 3)(x + 2)
8. (x - 5)(x + 4)
Exercise 2
1. (x - 7)(x - 2)
5. (a - 6)(a - 6)
Exercise 3
1. (x - 5)(x + 4)
5. ((x - 10)(x + 2)
Exercise 4
1. (x + 1)(x + 5)
5. (x + 2)(x + 7)
9. (x - 6)(x + 2)
Unitec: Document1
Unitec: Document1