GCSE(F), GCSE(H),

Factorising a quadratic expression, such as *x*^{2} + 6*x* + 8, means putting the brackets back in, to obtain an expression with two sets of brackets: (*x* + 4)(*x* + 2).

The answer will be in the form (*x* + *a*)(*x* + *b*), where *a* and *b* represent the missing numbers.

For the expression *x*^{2} + 8*x* + 12, the number 12 must be factors of 12 as *ab* = 12: possible factors are 1, 12 or 2, 6 or 3, 4.

The number 8 is obtained from adding the two *x* terms; *a* + *b* = 8.

So *a* and *b* are factors of 12 and add to 8. The pair of numbers that can do that are 6, 2.

*x*^{2} + 8*x* + 12 = (*x* + 6)(*x* + 2).

Negative signs have to be taken into account when determining the factors.

1. Factorise *x*^{2} + 13*x* + 42.

Answer: (*x* + 6)(*x* + 7)

factors of 42 | factors added |

(1, 42) | 43 |

(2, 21) | 23 |

(3, 14) | 17 |

(6, 7) | 13 |

The factors are therefore 6 and 7.

2. Factorise *a*^{2} - 8*a* + 15.

Answer: (*a* - 3)(*a* - 5)

factors of 15 | factors added |

(1, 15) | 16 |

(3, 5) | 8 |

(-3, -5) | -8 |

(-1, -15) | -16 |

The factors are -3 and -5.

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