### Factorising quadratics of the form x^{2} + bx + c

x^{2} + 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 x

^{2}+ 7x + 6#### Example 2. Factorise x

^{2}− 11x + 30#### Example 3. Factorise x

^{2}+ 2x − 8#### Example 4. Factorise x

^{2}− 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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