Lesson Guide
Explanation
Factorising undoes expanding. To factorise x squared plus seven x plus twelve, you need two numbers that multiply to twelve and add to seven. List the pairs that multiply to twelve: one and twelve, two and six, three and four. Only three and four add to seven. So the answer is x plus three, times x plus four. In the area model, the corner boxes are x squared and twelve, and the other two boxes, three x and four x, add up to the middle term, seven x.
Your Turn
Factorise x2 + 9x + 20. Check: 4 × 5 = 20 and 4 + 5 = 9, so (x + 4)(x + 5).
Watch Out
2 and 6 multiply to 12 but add to 8. The pair must do BOTH jobs: multiply to c and add to b.
Explanation
The sign of c tells you what to do. If c is positive, the two numbers have the same sign, and it is the sign of b. For x squared minus eight x plus fifteen, you need two numbers that multiply to positive fifteen and add to negative eight: negative three and negative five. If c is negative, the numbers have opposite signs, and the bigger one takes the sign of b. For x squared plus two x minus twenty-four, six and negative four multiply to negative twenty-four and add to positive two. For x squared minus three x minus ten, it is negative five and positive two.
Your Turn
Factorise x2 − x − 12. Check: −4 × 3 = −12 and −4 + 3 = −1, so (x − 4)(x + 3).
Watch Out
x2 + 2x − 24 is (x + 6)(x − 4), not (x − 6)(x + 4). The swapped signs give a middle term of −2x: expand to check.
Explanation
Before you look for two numbers, look for a common factor. Two x squared minus ten x minus twenty-eight has a two in every term, so take it out: two times x squared minus five x minus fourteen. Now the bracket is a monic quadratic. You need two numbers that multiply to negative fourteen and add to negative five: negative seven and positive two. So the answer is two, times x minus seven, times x plus two. Keep the two at the front: it is part of the answer.
Your Turn
Factorise 3x2 + 3x − 36 completely. Check: 3(x2 + x − 12) = 3(x + 4)(x − 3).
Watch Out
Do not drop the common factor at the end. (x − 7)(x + 2) is only x2 − 5x − 14, half of the original.
Explanation
When there is no x term, b is zero. For x squared minus forty-nine, you need two numbers that multiply to negative forty-nine and add to zero. They must be opposites: seven and negative seven. So x squared minus forty-nine is x plus seven, times x minus seven. That is the difference of two squares again, now run backwards. It works for any square: four x squared minus twenty-five is two x plus five, times two x minus five.
Your Turn
Factorise x2 − 81. Check: (x + 9)(x − 9).
Watch Out
x2 + 49 does not factorise: no two numbers multiply to +49 and add to 0. You need a minus sign between the squares.
Explanation
Sometimes the two numbers are the same. For x squared plus ten x plus twenty-five, five and five multiply to twenty-five and add to ten, so the answer is x plus five, all squared. Here is a quick test: halve b and square it. If you get c, it is a perfect square. Half of negative fourteen is negative seven, and negative seven squared is forty-nine, so x squared minus fourteen x plus forty-nine is x minus seven, all squared. Quiz and exercises now.
Your Turn
Factorise x2 − 6x + 9. Check: half of −6 is −3 and (−3)2 = 9, so (x − 3)2.
Watch Out
x2 − 12x + 36 is (x − 6)2, not (x + 6)(x − 6). That one expands to x2 − 36, with no middle term.
📌 Key Points — this lesson in a nutshell
- To factorise x2 + bx + c, find two numbers that multiply to c and add to b. Then x2 + bx + c = (x + p)(x + q).
- Signs: if c is positive, both numbers have the sign of b. If c is negative, the signs are opposite and the larger number takes the sign of b.
- Always take out a common factor first: 2x2 − 10x − 28 = 2(x2 − 5x − 14) = 2(x − 7)(x + 2).
- b = 0 gives a difference of two squares: x2 − k2 = (x + k)(x − k). If c = (half of b)2, it is a perfect square: x2 + 10x + 25 = (x + 5)2.
- Check by expanding: the middle term must come back exactly, sign and all.
Step 1: Introduction
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