Expanding Brackets
Expanding brackets turns up on nearly every GCSE Maths paper — single brackets like 3(x + 4) and doubles like (x + 2)(x + 5). This guide shows the method for each, the mistakes that cost marks, and exam-style questions with full worked solutions.
📋 Key Rules & Methods
Single Brackets
To expand a single bracket, multiply each term inside the bracket by the term outside.
Multiply every term inside by what's outside the bracket.
Double Brackets (FOIL Method)
For two brackets multiplied together, use FOIL: First, Outside, Inside, Last.
- First – multiply the first terms in each bracket
- Outside – multiply the outer terms
- Inside – multiply the inner terms
- Last – multiply the last terms in each bracket
Special Cases
- (x + a)² = x² + 2ax + a² – Square of a sum
- (x - a)² = x² - 2ax + a² – Square of a difference
- (x + a)(x - a) = x² - a² – Difference of two squares
✏️ Worked Examples
Expand: 4(2x - 3)
Answer: 8x - 12
Expand: -3(2x + 5)
Answer: -6x - 15
⚠️ Note: Both terms become negative because negative × positive = negative
Expand: (x + 3)(x + 5)
Answer: x² + 8x + 15
Expand: (x + 4)(x - 4)
Answer: x² - 16
💡 This pattern always gives x² - a² when you have (x + a)(x - a)
📝 Practice Questions
Try these questions yourself, then check your answers in the solutions section below.
✅ Answers & Worked Solutions
Q1: 5(x + 3) = 5x + 15
Q2: 3(2x - 7) = 6x - 21
Q3: -2(4x + 3) = -8x - 6
Remember: -2 × 3 = -6, not +6
Q4: 2(x + 4) + 3(x - 1) = 2x + 8 + 3x - 3 = 5x + 5
Q5: (x + 2)(x + 6) = x² + 6x + 2x + 12 = x² + 8x + 12
Q6: (x - 3)(x + 7) = x² + 7x - 3x - 21 = x² + 4x - 21
Q7: (x + 5)² = (x + 5)(x + 5) = x² + 5x + 5x + 25 = x² + 10x + 25
Q8: (2x + 3)(x - 4) = 2x² - 8x + 3x - 12 = 2x² - 5x - 12
Q9: (x + 3)(x - 3) = x² - 9 (difference of two squares)
Full working: x² - 3x + 3x - 9 = x² - 9
Q10: (3x - 2)² = (3x - 2)(3x - 2) = 9x² - 6x - 6x + 4 = 9x² - 12x + 4
⚠️ Common Mistakes to Avoid
Wrong: 3(x + 4) = 3x + 4 ❌
Correct: 3(x + 4) = 3x + 12 ✓
Remember to multiply EVERY term inside the bracket!
Wrong: -2(x + 3) = -2x + 6 ❌
Correct: -2(x + 3) = -2x - 6 ✓
Negative × positive = negative
Wrong: (x + 3)² = x² + 9 ❌
Correct: (x + 3)² = x² + 6x + 9 ✓
You must expand: (x + 3)(x + 3), don't just square each term!
Wrong: (x + 2)(x + 5) = x² + 10 ❌
Correct: (x + 2)(x + 5) = x² + 7x + 10 ✓
Don't forget the Outside and Inside terms!
❓ Frequently Asked Questions
What does expanding brackets mean in maths?
Expanding brackets means multiplying each term inside the bracket by the term outside. For example, 3(x + 4) expands to 3x + 12 because you multiply both x and 4 by 3. It's the opposite of factorising.
What is the FOIL method?
FOIL stands for First, Outside, Inside, Last. It's a systematic method for expanding two brackets multiplied together. Multiply the First terms, then Outside terms, then Inside terms, then Last terms, and add all the results together.
How do you expand brackets with a negative outside?
When there's a negative number outside the bracket, multiply every term inside by that negative number. For example, -2(x + 3) = -2x - 6. The key rule is: negative × positive = negative, and negative × negative = positive.
What is the difference between expanding and simplifying?
Expanding means multiplying out brackets. Simplifying means collecting like terms to make the expression shorter. Often you need to do both: first expand the brackets, then simplify by combining like terms.
How do you expand (x + a)²?
Write it as (x + a)(x + a) and use FOIL, or use the formula: (x + a)² = x² + 2ax + a². For example, (x + 3)² = x² + 6x + 9. Don't make the common mistake of thinking it equals x² + a²!
🎯 Practice More with Interactive Quizzes
Test your expanding brackets skills with our free interactive quiz and earn XP!
Start Practice Quiz →🔗 Related Topics
- Factorising – The reverse of expanding brackets
- Solving Equations – Uses expanding when equations have brackets
- Quadratic Equations – Requires expanding double brackets