Answers − Factorising quadratics of the form x2 + bx + c
1. Factorise x2 + 6x + 8
Here b = 6 and c = 8.
The factors of c = 8 are (1, 8) and (2, 4)
Out of the above factors choose (2, 4) as it satisfies the conditions:
n + q = b | n × q = c |
2 + 4 = 6 | 2 × 4 = 8 |
So the factors are (x + 2)(x + 4)
To check work in the opposite direction:
2. Factorise x2 + 9x + 18
Here b = 9 and c = 18.
The factors of c = 18 are (1, 18), (2, 9) and (3, 6)
Out of the above factors choose (3, 6) as it satisfies the conditions:
n + q = b | n × q = c |
3 + 6 = 9 | 3 × 6 = 18 |
So the factors are (x + 3)(x + 6)
To check work in the opposite direction:
3. Factorise x2 + x − 56
Here b = 1 and c = − 56. |
The factors of c = 56 are (1, 56), (2, 28), (4, 14) and (7, 8) |
Out of the above factors choose (7, 8) as it satisfies the conditions: |
n − q = b | n × −q=−c | n > q |
8 − 7 = 1 | 8 × −7=−56 | 8 > 7 |
So the factors are (x + 8)(x − 7)
To verify work in the opposite direction:
4. Factorise x2 + 10x − 24
Here b = 10 and c = − 24. |
The factors of c = 24 are (1, 24), (2, 12), (3, 8) and (4, 6) |
Out of the above factors choose (2, 12) as it satisfies the conditions: |
n − q=b | n × −q=−c | n > q |
12 − 2=10 | 12 × −2=−24 | 12>2 |
So the factors are (x + 12)(x − 2)
To verify work in the opposite direction:
5. Factorise x2 − 16x + 60
Here b = − 16 and c = 60. |
The factors of c = 60 are (1, 60), (2, 30), (3, 20), (4, 15), (5, 12) and (6, 10) |
Out of the above factors choose (6, 10) as it satisfies the conditions: |
−n − q = −b | −n × −q = c |
−6 − 10 = −16 | −6 × −10 = 60 |
So the factors are (x − 6)(x − 10)
To check work in the opposite direction:
6. Factorise x2 − 15x + 36
Here b = − 15 and c = 36. |
The factors of c = 36 are (1, 36), (2, 18), (3, 12), (4, 9) and (6, 6) |
Out of the above factors choose (3, 12) as it satisfies the conditions: |
−n − q = −b | −n × −q = c |
−3 − 12 = −15 | −3 × −12 = 36 |
So the factors are (x − 3)(x − 12)
To check work in the opposite direction:
7. Factorise x2 − 8x − 48
Here b = − 8 and c = − 48. |
The factors of c = 48 are (1, 48), (2, 24), (3, 16), (4, 12) and (6, 8) |
Out of the above factors choose (4, 12) as it satisfies the conditions: |
n − q=−b | n × −q=−c | q > n |
4 −12=−8 | 4 × −12=−48 | 12>4 |
So the factors are (x + 4)(x − 12)
To verify work in the opposite direction:
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