### Factorising quadratics of the form ax^{2} + bx + c, where a is not equal to 1

ax^{2} + bx + c can be factorised to (mx + n)(px + q). The x^{2} term has the value a which is not equal to 1. The process is similar to factorising x^{2} + bx + c in Factorising Quadratics − 1, but this time finding the factors of both a and c and using a combination of these two factors to work out the x term.

Note: the inverse of factorising quadratics is Quadratic Expansion

Note: for a recap on Quadratic Expansion refer to Quadratic Expansion

#### Example 1. Factorise 2x

^{2}+ 12x + 10#### Example 2. Factorise 4x

^{2}− 11x + 6#### Example 3. Factorise 2x

^{2}+ 2x − 24#### Example 4. Factorise 3x

^{2}− 4x − 20

Here a = 2, b = 12 and c = 10

i. | Possible factors of a: (2, 1) |

ii. | Possible factors of c: (5, 2) and (10, 1) |

iii. | Work out the x term. As c is positive, select two factors of a and c that total to b. |

Factor of a with first factor of c:

iv. | As c is postive the signs of the factors are the same as b: |

v. | To check work in the opposite direction: |

Here a = 4, b = −11 and c = 6

i. | Possible factors of a: (2, 2) and (4, 1) |

ii. | Possible factors of c: (3, 2) and (6, 1) |

iii. | Work out the x term. As c is positive, select two factors of a and c that total to b. |

First factor of a with first factor of c:

First factor of a with second factor of c:

Second factor of a with first factor of c:

iv. | As c is postive the signs of the factors are the same as b: |

v. | To verify work in the opposite direction: |

Here a = 2, b = 2 and c = −24

i. | Possible factors of a: (2, 1) |

ii. | Possible factors of c: (6, 4), (8, 3), (12, 2) and (24, 1) |

iii. | Work out the x term. As c is negative, select two factors of a and c that differ by b. |

Factor of a with first factor of c:

iv. | As c is negative the signs of the factors are different: |

v. | To check work in the opposite direction: |

Here a = 3, b = −4 and c = −20

i. | Possible factors of a: (3, 1) |

ii. | Possible factors of c: (5, 4), (10, 2) and (20, 1) |

iii. | Work out the x term. As c is negative, select two factors of a and c that differ by b. |

Factor of a with first factor of c:

Factor of a with second factor of c:

iv. | As c is negative the signs of the factors are different: |

v. | To verify work in the opposite direction: |

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