A trinomial or quadratic expression takes the form **ax ^{2} + bx + c **where a, b and c are numbers.

Sometimes it is not possible to factorise this expression into two brackets but, when it is, we can follow this method.

In ax^{2} + bx + c we look for pairs of numbers which multiply to give ac and add to give b.

**Example**

Factorise the following quadratic expression.

3x^{2} - 2x - 8

**Answer**

3x^{2} - 2x - 8

Here we have a = 3, b = -2 and c = -8.

So we list the factor pairs of -24 and see which pair adds to give -2.

Factor Pairs | Sum of factors |
---|---|

1 and -24 | -23 |

-1 and 24 | 23 |

2 and -12 | -10 |

-2 and 12 | 10 |

3 and -8 | -5 |

-3 and 8 |
5 |

4 and -6 | -2 |

-4 and 6 | 2 |

We now split the middle term -2x using the chosen factor pair i.e. 4x and -6x

3x^{2 }+ 4x - 6x - 8

Consider the expression in pairs

3x^{2} + 4x - 6x - 8

Factorise each pair

x (3x + 4) - 2 (3x + 4)

Combine

(x - 2) (3x + 4)

This is the full factorisation

**(x - 2) (3x + 4)**