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.
Note: The inverse of factorising quadratics is Quadratic Expansion
Note: for a recap on Quadratic Expansion refer to Quadratic Expansion
Example 1. Factorise x2 + 7x + 6
Example 2. Factorise x2 − 11x + 30
Example 3. Factorise x2 + 2x − 8
Example 4. Factorise x2 − 5x − 24
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:
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:
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:
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:
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