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.

18x^{2} + 3x - 10

**Answer**

18x^{2} + 3x - 10

Here we have a = 18, b = 3 and c = -10.

So we list the factor pairs of -180 and see which pair adds to give 3.

The first two factor pairs below are too wildly out, so, rather than looking at the extreme factor pairs, we concentrate on ones that are closer together.

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

1 and -180 | -179 |

-1 and 180 | 179 |

18 and -10 | 8 |

-18 and 10 | -8 |

15 and -12 | 3 |

-15 and 12 |
-3 |

30 and -6 | 24 |

-30 and 6 | -24 |

We now split the middle term +3x using the chosen factor pair i.e. 15x and -12x

18x^{2 }+ 15x - 12x - 10

Consider the expression in pairs

18x^{2} + 15x - 12x - 10

Factorise each pair

3x (6x + 5) - 2 (6x + 5)

Combine

(3x - 2) (6x + 5)

This is the full factorisation

**(3x - 2) (6x + 5)**