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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