## Factorising quadratics with integer roots

The general form of a quadratic is

We are going to consider cases where

Consider

If we expand the brackets we get

This simplifies to

It can be seen from above that the sum of and is equal to the coefficient of x

and that the product of and is equal to the value of the constant.

So to factorise a quadratic first:

- Find all the pairs of integers, positive and negative, that multiply together to make the constant value
- See if one of the pairs if added instead can equal the coefficient of x
- If such a pair can be found use them to replace and in the formula
- If a pair can not be found then a different method is needed

Eamples

Factorise

First lets find the pairs of numbers that when multiplied together make +15.

1 | 15 |

-1 | -15 |

3 | 5 |

-3 | -5 |

Now see if any pair when added together we get the value +8

1 | 15 | 16 |

-1 | -15 | -16 |

3 | 5 | 8 |

-3 | -5 | -8 |

Since 3 & 5 multiply together to make 15 and add together to make 8 these are out and

So

(Note that I tried to be systematic when finding the factor pairs which equalled 15. Also note that it does not matter if you give the factorised expression as or

Factorise

The factor pairs

1 | 12 |

-1 | -12 |

2 | 6 |

-2 | -6 |

3 | 4 |

-3 | -4 |

Now total each pair

1 | 12 | 13 |

-1 | -12 | -13 |

2 | 6 | 8 |

-2 | -6 | -8 |

3 | 4 | 7 |

-3 | -4 | -7 |

Since -3 & -4 multiply together to make 12 and add together to make -7 these are out and

So

Factorise

The factor pairs

1 | -10 |

-1 | 10 |

2 | -5 |

-2 | 5 |

Now total each pair

1 | -10 | -9 |

-1 | 10 | 9 |

2 | -5 | -3 |

-2 | 5 | 3 |

Since -2 & 5 multiply together to make -10 and add together to make 3 these are out and

So

Factorise

The factor pairs

1 | -21 |

-1 | 21 |

3 | -7 |

-3 | 7 |

Now total each pair

1 | -21 | -20 |

-1 | 21 | 21 |

3 | -7 | -4 |

-3 | 7 | 4 |

Since 3 & -7 multiply together to make -21 and add together to make -4 these are out and

So

Factorise

The factor pairs

1 | -7 |

-1 | 7 |

Now total each pair

1 | -7 | -6 |

-1 | 7 | 6 |

Since no pair has a total of -2 that we required and because we have been systematic we know we have considered all possible pairs, we can assume that either this expression does not factorise or that a different method is required.