Every GCSE maths paper has a factorising question. The trick that cracks it makes sense the moment you watch the brackets work backwards.
Watch it backwards first
Start with brackets you already know how to expand: (x + 2)(x + 3). Multiply it out and you get x² + 3x + 2x + 6, which tidies up to x² + 5x + 6.
Now look at where the 5 and the 6 came from. 2 + 3 = 5, and 2 × 3 = 6. The middle number is the two numbers added. The last number is the two numbers multiplied.
So factorising is one question
Which two numbers multiply to the last number, and add to the middle one? Find them and you have the brackets.
Worked example: x² + 7x + 12
Don't guess. List the pairs that multiply to 12, in order:
- 1 and 12, which add to 13
- 2 and 6, which add to 8
- 3 and 4, which add to 7 ✓
3 and 4 add to 7, so x² + 7x + 12 = (x + 3)(x + 4). Listing the pairs in order is the habit that works on every question, not just the easy ones.
Check your answer
Expand your brackets again: (x + 3)(x + 4) = x² + 4x + 3x + 12 = x² + 7x + 12. If you get back to where you started, it's right. This video covers positive numbers only. Part 2 shows what happens when a minus sign turns up.
Factorise x² + 8x + 15.
Show the answer
3 × 5 = 15 and 3 + 5 = 8, so it's (x + 3)(x + 5).
Save this for your next test
- Multiply to the last number.
- Add to the middle number.
- List the factor pairs in order instead of guessing.
- Expand your answer to check it.
Read the video transcript
- Every GCSE maths paper has a question like this.
- And there's a trick that cracks it in ten seconds.
- First, watch it backwards.
- Here's x plus two, times x plus three.
- Multiply it out, and you get x squared, plus five x, plus six.
- Look. Two plus three is five. Two times three is six.
- So factorising is just one question: which two numbers multiply to the last number, and add to the middle one?
- Let's try one. x squared, plus seven x, plus twelve.
- Pairs that multiply to twelve: one and twelve. Two and six. Three and four.
- Which pair adds to seven? Three and four!
- So it's x plus three, times x plus four.
- Your turn. x squared, plus eight x, plus fifteen. Three… two… one…
- Three and five! x plus three, times x plus five.
- Next time: what happens when the numbers go negative.
- Want a tutor who makes maths click like this? Book a free trial lesson at bloomeducationonline.com.
Want maths to click like this?
Every Bloom lesson works the same way: one idea at a time, shown before it's explained, then practised until it sticks. Book a free assessment and see it applied to your child's own work.
