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

#### 1. Factorise x^{2} + 6x + 8

Here b = 6 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 |

2 + 4 = 6 | 2 × 4 = 8 |

So the factors are (x + 2)(x + 4)

To check work in the opposite direction:

#### 2. Factorise x^{2} + 9x + 18

Here b = 9 and c = 18.

The factors of c = 18 are (1, 18), (2, 9) and (3, 6)

Out of the above factors choose (3, 6) as it satisfies the conditions:

n + q = b | n × q = c |

3 + 6 = 9 | 3 × 6 = 18 |

So the factors are (x + 3)(x + 6)

To check work in the opposite direction:

#### 3. Factorise x^{2} + x − 56

Here b = 1 and c = − 56. |

The factors of c = 56 are (1, 56), (2, 28), (4, 14) and (7, 8) |

Out of the above factors choose (7, 8) as it satisfies the conditions: |

n − q = b | n × −q=−c | n > q |

8 − 7 = 1 | 8 × −7=−56 | 8 > 7 |

So the factors are (x + 8)(x − 7)

To verify work in the opposite direction:

#### 4. Factorise x^{2} + 10x − 24

Here b = 10 and c = − 24. |

The factors of c = 24 are (1, 24), (2, 12), (3, 8) and (4, 6) |

Out of the above factors choose (2, 12) as it satisfies the conditions: |

n − q=b | n × −q=−c | n > q |

12 − 2=10 | 12 × −2=−24 | 12>2 |

So the factors are (x + 12)(x − 2)

To verify work in the opposite direction:

#### 5. Factorise x^{2} − 16x + 60

Here b = − 16 and c = 60. |

The factors of c = 60 are (1, 60), (2, 30), (3, 20), (4, 15), (5, 12) and (6, 10) |

Out of the above factors choose (6, 10) as it satisfies the conditions: |

−n − q = −b | −n × −q = c |

−6 − 10 = −16 | −6 × −10 = 60 |

So the factors are (x − 6)(x − 10)

To check work in the opposite direction:

#### 6. Factorise x^{2} − 15x + 36

Here b = − 15 and c = 36. |

The factors of c = 36 are (1, 36), (2, 18), (3, 12), (4, 9) and (6, 6) |

Out of the above factors choose (3, 12) as it satisfies the conditions: |

−n − q = −b | −n × −q = c |

−3 − 12 = −15 | −3 × −12 = 36 |

So the factors are (x − 3)(x − 12)

To check work in the opposite direction:

#### 7. Factorise x^{2} − 8x − 48

Here b = − 8 and c = − 48. |

The factors of c = 48 are (1, 48), (2, 24), (3, 16), (4, 12) and (6, 8) |

Out of the above factors choose (4, 12) as it satisfies the conditions: |

n − q=−b | n × −q=−c | q > n |

4 −12=−8 | 4 × −12=−48 | 12>4 |

So the factors are (x + 4)(x − 12)

To verify work in the opposite direction:

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