Factorise quadratics, differences of squares and four terms
Factor a quadratic by splitting its middle term · Group four terms to reveal a common bracket · Recognise repeated difference-of-squares structure
Exam scope and assessment notes
Extended content; not a Core requirement. Historical study of Cambridge IGCSE Mathematics (US) 0444, 2025 Version 2. The final assessment was June 2025; the syllabus was available only to schools in Arizona, USA. This is not a current exam-entry option. Original self-check teaching and explained practice are not Cambridge papers, official marks or predicted grades. Linked teaching foundations do not establish full syllabus mastery. Calibrated cumulative assessment, reliable drawn-work review and qualified mathematics-teacher review remain unfinished.
Before you begin
- Common-factor factorisation
- Two-bracket expansion
Work on paper. Try each question before revealing help, and explain why your method works.
Quick readiness check
What should you establish before applying the method in this lesson?
What you will learn
- Factor a quadratic by splitting its middle term
- Group four terms to reveal a common bracket
- Recognise repeated difference-of-squares structure
Split and group
For ax²+bx+c, look for two numbers with product ac and sum b. For 6x²+7x−3, use 9 and −2: 3x(2x+3)−1(2x+3) = (3x−1)(2x+3). Verify by expanding.
Four terms may group into matching brackets: ab−4a+3b−12 = a(b−4)+3(b−4) = (a+3)(b−4). Choose grouping to reveal structure rather than removing a different unrelated factor from each term.
Difference of squares
A²−B² = (A−B)(A+B). For x⁴−y⁴, see squares of x² and y²: (x²−y²)(x²+y²), then factor the first bracket again as (x−y)(x+y). x²+y² has no analogous real linear factorisation using this rule.
Equivalent forms can reveal roots, factors or signs. Do not treat a factorised expression as an equation unless an equality condition is actually given.
Sources for this lesson
Pause and explain
Does x²+y² equal (x−y)(x+y)?
Worked example
Factorise 6x²+7x−3.
Show the worked solution
- Split 7x as 9x−2x.
- Group: 3x(2x+3)−(2x+3).
- Factor the shared bracket.
Answer (3x−1)(2x+3).
Try a topic test
Choose Easy, Normal or Difficult. Save your answers and open worked review after finishing.
Test this topic →From guided practice to a new situation
Write your answer and reasoning first. Use a hint only if you are stuck. The model solution is for self-checking; your written work is not automatically marked.
Factorise x²+5x+6.
Give me a hint
Find a pair with product 6 and sum 5.
Compare my reasoning
- 2 and 3 satisfy both conditions.
- Expand (x+2)(x+3) to check.
(x+2)(x+3).
Look for these in your work
- Check product and sum
- Verify by expansion
Factorise ab−4a+3b−12.
Give me a hint
Group the first two and last two terms.
Compare my reasoning
- a(b−4)+3(b−4).
- Extract the common bracket.
(a+3)(b−4).
Look for these in your work
- Find a matching bracket
- Retain all four terms
Factorise x⁴−y⁴ as far as the real difference-of-squares rule allows.
Give me a hint
First treat x² and y² as the two squared quantities.
Compare my reasoning
- (x²−y²)(x²+y²).
- Factor x²−y² once more; retain the sum-of-squares factor.
(x−y)(x+y)(x²+y²).
Look for these in your work
- Recognise structure twice
- Do not misuse the rule for a sum
Common mistakes
- Check both product and sum when splitting a middle term.
- The difference-of-squares rule needs subtraction.
What can you explain now?
Does x²+y² equal (x−y)(x+y)?
Compare with the explanation
No; that product equals x²−y².
Expansion shows the sign of the last term.
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Make it stick
Does x²+y² equal (x−y)(x+y)?
If you needed the solution, close it and solve the problem again from a blank page. A correct answer is stronger when you can explain the reason for each step.