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One idea first
A quadratic like x^2 + 7x + 12 can be factorised by finding two numbers that multiply to 12 and add to 7. Those numbers are 3 and 4, so the expression becomes (x + 3)(x + 4).
Why this matters: Quadratics appear in algebra, graphs, area problems, and later differentiation. Secure factorising makes higher-mark algebra feel less random.
Quick hook
GCSE Maths: factorising quadratics without panic: the tiny version is simpler than it looks. One move first, then the deeper stuff.
Brain shortcut
Treat Maths like a messy group chat: find the message that actually matters, mute the noise, then reply clearly.
Tiny win
Before solving, say the task in one sentence. That is the anti-panic button.
Deep bit
Quadratics appear in algebra, graphs, area problems, and later differentiation. Secure factorising makes higher-mark algebra feel less random.
Rapid check: A quadratic like x^2 + 7x + 12 can be factorised by finding two numbers that multiply to 12 and add to 7. Those numbers are 3 and 4, so the expression becomes (x + 3)(x + 4).
Deep explanation
Factorising reverses expanding brackets. When you expand (x + a)(x + b), the outside and inside terms combine to make (a + b)x, and the constants multiply to make ab. That means x^2 + 7x + 12 is asking for two hidden constants whose sum is 7 and product is 12. Listing factor pairs keeps the load low: 1 and 12 add to 13, 2 and 6 add to 8, 3 and 4 add to 7. Once the pair works, place them in brackets and check by expanding back. This check matters because sign errors are the most common way students lose marks even when they know the method.
Visual model
Imagine the middle term as a bridge between two brackets. The product chooses the constants; the sum checks whether the bridge is the right size.
- 1. Write the product target under the constant term.
- 2. List factor pairs and compare their sum with the x coefficient.
- 3. Place the pair into brackets, then expand back as a check.
Worked example
Factorise x^2 + 8x + 15.
Step 1: Find the product target
The constant is 15, so list factor pairs of 15.
Why: The two bracket constants must multiply to the final constant.
Step 2: Check the sum
1 and 15 add to 16. 3 and 5 add to 8.
Why: The middle coefficient is 8, so 3 and 5 are the matching pair.
Step 3: Write and check
x^2 + 8x + 15 = (x + 3)(x + 5). Expanding gives x^2 + 5x + 3x + 15.
Why: The check proves both the middle term and constant are correct.
Final answer: (x + 3)(x + 5)
Predict the next step
For x^2 + 10x + 24, which pair should go in the brackets?
Show feedback
4 and 6 multiply to 24 and add to 10, so the factors are (x + 4)(x + 6).
Practice ladder
Factorise x^2 + 5x + 6.
Show hints and explanation
- - List factor pairs of 6.
- - Which pair adds to 5?
Answer: (x + 2)(x + 3)
2 and 3 multiply to 6 and add to 5.
Factorise x^2 + 11x + 30.
Show hints and explanation
- - Try factor pairs of 30.
- - Check the pair against the x coefficient.
Answer: (x + 5)(x + 6)
5 and 6 multiply to 30 and add to 11.
Factorise x^2 - x - 12.
Show hints and explanation
- - A negative product means one sign is positive and one is negative.
- - Which pair differs by 1 and gives a negative sum?
Answer: (x - 4)(x + 3)
-4 and 3 multiply to -12 and add to -1.
The area of a rectangle is x^2 + 7x + 10. One side is x + 5. Find the other side.
Show hints and explanation
- - Factorise the area expression.
- - Match one bracket with the side you already know.
Answer: x + 2
Factorise x^2 + 7x + 10 as (x + 5)(x + 2), so the missing side is x + 2.
Flashcard reinforcement
What two checks does a quadratic factor pair need?
The pair must multiply to the constant and add to the x coefficient.
Product then sum.
Why expand your brackets after factorising?
It catches wrong signs and product-only mistakes.
Check by reversing.
What does a negative product tell you?
One bracket sign is positive and one is negative.
Negative product means split signs.
Misconception fixer
Only checking the product
The constant is visually obvious, so students stop too early.
Fix: Always say the sum out loud before writing brackets.
Losing the negative sign
Students treat the numbers as positive factor pairs and add signs later.
Fix: Choose signs while checking the sum.
Assessment technique
Marks come from visible method
Show the factor pair check before the final brackets when the question is worth more than one mark. If the expression has a negative constant, make the sign decision explicit so the examiner can see your reasoning.
UK GCSE questions usually reward the correct factorisation and may reward working when the expression is embedded in a context.
Home-study pack
- Expand two bracket pairs.
- Factorise three easy quadratics.
- Correct one sign-error example.
- Write a one-line explanation of product then sum.
The learner is practising the core GCSE algebra habit of checking both product and sum before writing bracket factors.
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