Expand two brackets and the square of a binomial
Multiply every term in one bracket by every term in the other · Use the cross term in a binomial square · Check equivalent expressions
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
- Simple expansion
- Algebraic powers
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
- Multiply every term in one bracket by every term in the other
- Use the cross term in a binomial square
- Check equivalent expressions
Four products
(a+b)(c+d) = ac+ad+bc+bd. For (a+b)² the two middle products are both ab, giving a²+2ab+b². For (a−b)² they are both −ab, giving a²−2ab+b².
Different operations
Squaring a sum is not adding squares: (x+3)² = x²+6x+9. Multiplying conjugates is different: (x+3)(x−3) = x²−9 because the cross terms cancel. Expand to verify before combining like terms.
Sources for this lesson
Pause and explain
What is (x+3)²?
Worked example
Expand (2x−3)².
Show the worked solution
- Write (2x−3)(2x−3).
- Products are 4x², −6x, −6x and 9; combine the middle pair.
Answer 4x²−12x+9.
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.
Expand (x+2)(x+5).
Give me a hint
Make all four products.
Compare my reasoning
- x²+5x+2x+10.
- Collect to x²+7x+10.
x²+7x+10.
Look for these in your work
- Multiply every term
- Collect like terms
Expand (3y−2)².
Give me a hint
Square both terms and include twice their product.
Compare my reasoning
- 9y²−6y−6y+4.
- Combine the cross terms.
9y²−12y+4.
Look for these in your work
- Retain the negative cross term
- Check the squared constant
A square side grows from x to x+4, with x > 0. Form the increase in area.
Give me a hint
Subtract the old area from the expanded new area.
Compare my reasoning
- New area (x+4)² = x²+8x+16.
- Subtract x² to get 8x+16.
Area increase 8x+16.
Look for these in your work
- Distinguish side increase from area increase
- Use the full expansion
Common mistakes
- Keep the cross term in a squared binomial.
- A square and a conjugate product are different.
What can you explain now?
What is (x+3)²?
Compare with the explanation
x²+6x+9.
The two cross products contribute 3x+3x.
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Make it stick
What is (x+3)²?
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.