Expanding & Simplifying Single Brackets
GCSE Mathematics📦 What is Expanding Brackets?
Expanding brackets (or multiplying out) means removing the brackets by multiplying everything inside the brackets by the term outside. It's like distributing items from a box to everyone - each term inside gets multiplied by the term outside.
Example: 3(x + 4) = 3 × x + 3 × 4 = 3x + 12
🎨 Visualizing Expansion
↓ Multiply each term inside by 3 ↓
📚 Types of Brackets
➕ Positive Number Outside
Multiply the positive number by each term inside.
Example 1
4(x + 3)
= 4 × x + 4 × 3
= 4x + 12
Example 2
5(2x + 4)
= 5 × 2x + 5 × 4
= 10x + 20
Example 3
3(3x + 2y)
= 3 × 3x + 3 × 2y
= 9x + 6y
➖ Negative Number Outside
Be careful with signs! Negative × positive = negative, Negative × negative = positive.
Example 1
-3(x + 4)
= (-3) × x + (-3) × 4
= -3x - 12
Example 2
-2(3x - 5)
= (-2) × 3x + (-2) × (-5)
= -6x + 10
Example 3
-4(2x + 3y)
= (-4) × 2x + (-4) × 3y
= -8x - 12y
🔤 Variable Outside
Multiply the variable by each term inside (remember laws of indices).
Example 1
x(x + 5)
= x × x + x × 5
= x² + 5x
Example 2
2x(3x - 4)
= 2x × 3x + 2x × (-4)
= 6x² - 8x
Example 3
3y(2y + 4z)
= 3y × 2y + 3y × 4z
= 6y² + 12yz
🔢 Both Number & Variable
Multiply both the number and variable parts separately.
Example 1
3x(2x + 5)
= 3x × 2x + 3x × 5
= 6x² + 15x
Example 2
-2x(3x - 4y)
= (-2x) × 3x + (-2x) × (-4y)
= -6x² + 8xy
Example 3
4xy(2x + 3y)
= 4xy × 2x + 4xy × 3y
= 8x²y + 12xy²
🧮 Interactive Bracket Expander
Enter your own expression and see it expand step-by-step!
✏️ Practice Expanding Brackets
Expand and simplify: 3(x + 4)
🔄 Expanding and Then Simplifying
Sometimes after expanding, you need to collect like terms:
Example 1
2(x + 3) + 3(x + 2)
Step 1: Expand both brackets
= 2x + 6 + 3x + 6
Step 2: Collect like terms
= (2x + 3x) + (6 + 6)
= 5x + 12
Example 2
4(x - 2) - 2(x + 1)
Step 1: Expand both brackets
= 4x - 8 - 2x - 2
Step 2: Collect like terms
= (4x - 2x) + (-8 - 2)
= 2x - 10
Example 3
3x(2x + 1) - 2x(x - 4)
Step 1: Expand both brackets
= 6x² + 3x - 2x² + 8x
Step 2: Collect like terms
= (6x² - 2x²) + (3x + 8x)
= 4x² + 11x
⚠️ Common Mistakes
❌ Wrong: Only multiplying the first term
3(x + 4) = 3x + 4 ❌
Correct: 3(x + 4) = 3x + 12
✅ Correct: Multiply every term inside
3(x + 4) = 3 × x + 3 × 4 = 3x + 12
❌ Wrong: Sign errors with negatives
-2(x - 3) = -2x - 6 ❌
Correct: -2(x - 3) = -2x + 6
⚠️ Remember: Negative × Negative = Positive
(-3) × (-4) = +12
❌ Wrong: Forgetting to multiply coefficients
2x(3x + 4) = 2x² + 8x ❌
Correct: 2x(3x + 4) = 6x² + 8x
🌍 Where Bracket Expansion is Used
📏 Area Calculation
Rectangle with length (x+3) and width 4:
Area = 4(x + 3) = 4x + 12
💰 Perimeter Problems
Rectangle with sides (x+2) and (x+1):
Perimeter = 2(x+2) + 2(x+1)
= 2x+4 + 2x+2 = 4x+6
📊 Cost Calculations
Buying x items at price (p+2) each:
Total = x(p+2) = xp + 2x