Expanding Double Brackets
GCSE Mathematics📦 What are Double Brackets?
Expanding double brackets means multiplying two binomial expressions together. Every term in the first bracket must be multiplied by every term in the second bracket. The common method is FOIL: First, Outer, Inner, Last.
Example: (x + 3)(x + 4) = x² + 4x + 3x + 12 = x² + 7x + 12
🌈 The FOIL Method
x × x = x²
x × 4 = 4x
3 × x = 3x
3 × 4 = 12
📚 Types of Double Brackets
➕ Both Brackets Positive
All terms are positive - straightforward multiplication.
Example 1
(x + 2)(x + 5)
= x×x + x×5 + 2×x + 2×5
= x² + 5x + 2x + 10
= x² + 7x + 10
Example 2
(x + 3)(x + 7)
= x² + 7x + 3x + 21
= x² + 10x + 21
Example 3
(x + 4)(x + 6)
= x² + 6x + 4x + 24
= x² + 10x + 24
➖ With Negative Signs
Be careful with signs! Negative × Negative = Positive.
Example 1
(x - 2)(x + 5)
= x×x + x×5 + (-2)×x + (-2)×5
= x² + 5x - 2x - 10
= x² + 3x - 10
Example 2
(x + 3)(x - 4)
= x² - 4x + 3x - 12
= x² - x - 12
Example 3
(x - 3)(x - 5)
= x² - 5x - 3x + 15
= x² - 8x + 15
🔢 With Coefficients
Multiply coefficients together and use index laws for variables.
Example 1
(2x + 3)(x + 4)
= 2x×x + 2x×4 + 3×x + 3×4
= 2x² + 8x + 3x + 12
= 2x² + 11x + 12
Example 2
(3x - 2)(2x + 5)
= 3x×2x + 3x×5 + (-2)×2x + (-2)×5
= 6x² + 15x - 4x - 10
= 6x² + 11x - 10
Example 3
(4x - 1)(3x - 2)
= 4x×3x + 4x×(-2) + (-1)×3x + (-1)×(-2)
= 12x² - 8x - 3x + 2
= 12x² - 11x + 2
⬆️ Perfect Squares
When both brackets are the same: (a + b)² = a² + 2ab + b²
Example 1
(x + 3)²
= (x + 3)(x + 3)
= x² + 3x + 3x + 9
= x² + 6x + 9
Example 2
(2x - 5)²
= (2x - 5)(2x - 5)
= 4x² - 10x - 10x + 25
= 4x² - 20x + 25
Formula
(a + b)² = a² + 2ab + b²
(a - b)² = a² - 2ab + b²
🔄 Difference of Squares
When brackets are (a + b)(a - b): The middle terms cancel!
Example 1
(x + 3)(x - 3)
= x² - 3x + 3x - 9
= x² - 9
Example 2
(2x + 5)(2x - 5)
= 4x² - 10x + 10x - 25
= 4x² - 25
Formula
(a + b)(a - b) = a² - b²
🧮 Interactive Double Bracket Expander
Enter your own double brackets and see them expand step-by-step!
✏️ Practice Expanding Double Brackets
Expand and simplify: (x + 3)(x + 4)
🌟 Challenge Questions
Expand and simplify: (2x + 3)(3x - 4)
Expand and simplify: (3x - 2)(2x - 5)
Expand and simplify: (4x + 3)²
⚠️ Common Mistakes
❌ Wrong: Only multiplying First and Last
(x + 3)(x + 4) = x² + 12 ❌
Correct: x² + 4x + 3x + 12 = x² + 7x + 12
✅ Correct: Use FOIL
First: x × x = x²
Outer: x × 4 = 4x
Inner: 3 × x = 3x
Last: 3 × 4 = 12
❌ Wrong: Sign errors
(x - 3)(x + 4) = x² + 4x - 3x - 12 ❌
Wait, that's actually correct! But some forget: (-3)×4 = -12
⚠️ Forgetting to simplify
Always combine like terms (the Outer and Inner terms)
❌ Wrong: Messing up coefficients
(2x + 3)(3x + 4) = 2x×3x + 2x×4 + 3×3x + 3×4
= 6x² + 8x + 9x + 12 = 6x² + 17x + 12
🌍 Where Double Brackets Are Used
📐 Area of Rectangle
Length = (x + 5), Width = (x + 3)
Area = (x + 5)(x + 3)
= x² + 3x + 5x + 15
= x² + 8x + 15
🏢 Building Design
Floor dimensions: (2x + 10) by (3x + 5)
Area = 6x² + 10x + 30x + 50
= 6x² + 40x + 50
💰 Profit Calculation
Profit = (price - cost) × quantity
If price = (x + 20), quantity = (x + 100)
Profit expands to quadratic