### Factorising quadratics of the form x2 + bx + c

x2 + bx + c can be factorised to (x + n)(x + q). There are four rules depending on the signs of b and c.

• #### Example 1.   Factorise x2 + 7x + 6

• Here b and c are both positive. Find two numbers n and q so that:

 n + q = b n × q = c

In this example b = 7 and c = 6.
The factors of c = 6 are (1, 6) and (2, 3)
Out of the above factors choose (1, 6) as it satisfies the conditions:

 n + q = b n × q = c 1 + 6 = 7 1 × 6 = 6

So the factors are (x + 1)(x + 6)

To check work in the opposite direction:

• #### Example 2.   Factorise x2 − 11x + 30

• Here b is negative and c is positive. Find two numbers n and q so that:

 −n − q = −b −n × −q = c
 In this example b = −11 and c = 30. The factors of c = 30 are (1, 30), (2, 15), (3, 10) and (5, 6) Out of the above factors choose (5, 6) as it satisfies the conditions:
 −n − q = −b −n × −q = c −5 − 6 = −11 −5 × −6 = 30

So the factors are (x − 5)(x − 6)

To check work in the opposite direction:

• #### Example 3.   Factorise x2 + 2x − 8

• Here b is positive and c is negative. Find two numbers n and q so that:

 n − q = b n × −q = −c n > q
 In this example b = 2 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 n > q 4 − 2 = 2 4 × −2 = −8 4 > 2

So the factors are (x + 4)(x − 2)

To verify work in the opposite direction:

• #### Example 4.   Factorise x2 − 5x − 24

• Here b is negative and c is negative. Find two numbers n and q so that:

 n − q =−b n × −q =−c q > n
 In this example b = −5 and c = −24. The factors of c = 24 are (1, 24), (2, 12), (3, 8) and (4, 6) Out of the above factors choose (3, 8) as it satisfies the conditions:
 n − q=−b n × −q=−c q > n 3 − 8=−5 3 × −8=−24 8 > 3

So the factors are (x + 3)(x − 8)

To verify work in the opposite direction:

to: