1.2 Expanding Brackets (GCSE)
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Learning Objectives
- Understand the distributive property of multiplication over addition
- Expand single and double brackets
- Simplify expressions by expanding and collecting like terms
- Solve equations involving expanded brackets
- Apply bracket expansion to real-world problems
Key Concepts
Introduction to Expanding Brackets
Expanding brackets involves applying the distributive property, where each term inside the bracket is multiplied by the term outside.
For example, to expand 2(x + 3), we multiply 2 by both x and 3: 2(x + 3) = 2x + 6.
This process is fundamental for simplifying algebraic expressions and solving equations.
Expanding Single Brackets
When you have a single bracket, distribute the term outside the bracket to each term inside.
For example, a(b + c) = ab + ac.
Example 1
Expand 3(x + 4)
Solution:
Using the distributive property: 3(x + 4) = 3x + 12
Example 2
Expand 2(a - 5)
Solution:
Using the distributive property: 2(a - 5) = 2a - 10
Expanding Double Brackets
When expanding double brackets, use the FOIL method: First, Outer, Inner, Last.
For example, (x + 2)(x + 3) expands to: x² + 3x + 2x + 6.
Example 1
Expand (x + 2)(x + 3)
Solution:
Using FOIL: (x + 2)(x + 3) = x² + 3x + 2x + 6 = x² + 5x + 6
Example 2
Expand (a - 1)(a + 4)
Solution:
Using FOIL: (a - 1)(a + 4) = a² + 4a - a - 4 = a² + 3a - 4
Expanding and Collecting Like Terms
After expanding brackets, you may need to collect like terms to simplify the expression.
For example, after expanding (x + 3)(x + 4), we can collect the x terms and the constant terms.
Example 1
Expand and simplify (x + 3)(x + 5)
Solution:
Expanding: (x + 3)(x + 5) = x² + 5x + 3x + 15 = x² + 8x + 15
Example 2
Expand and simplify (2x - 1)(x + 4)
Solution:
Expanding: (2x - 1)(x + 4) = 2x² + 8x - x - 4 = 2x² + 7x - 4
Solving Equations Involving Brackets
You may be asked to solve equations that contain brackets. The first step is to expand the brackets, then solve the resulting equation.
For example, to solve 2(x + 3) = 14, expand the brackets first to get 2x + 6 = 14, then solve for x.
Example 1
Solve 3(x + 4) = 18
Solution:
Expanding: 3x + 12 = 18. Subtract 12 from both sides: 3x = 6. Dividing by 3: x = 2.
Example 2
Solve 4(x - 2) = 12
Solution:
Expanding: 4x - 8 = 12. Add 8 to both sides: 4x = 20. Dividing by 4: x = 5.
Advanced Bracket Expansion
Sometimes you'll encounter more complex expressions with multiple brackets or nested brackets.
Always work from the inside out, expanding the innermost brackets first.
Example 1
Expand (x + 2)(x + 3)(x + 4)
Solution:
Step 1: Expand the first two brackets: (x + 2)(x + 3) = x² + 5x + 6 Step 2: Multiply by the third bracket: (x² + 5x + 6)(x + 4) = x³ + 4x² + 5x² + 20x + 6x + 24 = x³ + 9x² + 26x + 24
Example 2
Expand (2x + 1)²
Solution:
Using the formula (a + b)² = a² + 2ab + b²: (2x + 1)² = (2x)² + 2(2x)(1) + 1² = 4x² + 4x + 1
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