Expanding Brackets Quizzes

Introduction

Expanding brackets is a fundamental algebra skill in GCSE Maths. It involves removing brackets in expressions by multiplying each term inside the bracket by the factor outside. This skill is essential for simplifying expressions, solving equations, factorising, and manipulating algebraic formulas. Mastering bracket expansion helps students work confidently with both numbers and variables in a variety of contexts.

For example, $$3(x + 4)$$ expands to $$3x + 12$$. Expanding brackets correctly ensures accuracy when performing algebraic operations and solving problems efficiently.

Core Concepts

Single Brackets

When an expression has a single bracket, multiply each term inside by the term outside:

Formula: $$a(b + c) = ab + ac$$

Examples:

• $$5(x + 2) = 5x + 10$$
• $$-3(y - 4) = -3y + 12$$
• $$2(3x + 5) = 6x + 10$$

Double Brackets (Binomial × Binomial)

When multiplying two binomials, multiply each term in the first bracket by each term in the second:

Formula: $$(a + b)(c + d) = ac + ad + bc + bd$$

Examples:

• $$ (x + 3)(x + 5) = x^2 + 5x + 3x + 15 = x^2 + 8x + 15 $$
• $$ (2x - 3)(x + 4) = 2x^2 + 8x - 3x - 12 = 2x^2 + 5x - 12 $$

Brackets with Negative Signs

Pay attention to negative signs when expanding:

• $$-(x + 5) = -x - 5$$
• $$-3(x - 2) = -3x + 6$$
• $$- (2x - 7) = -2x + 7$$

Expanding with Coefficients and Indices

When the terms inside brackets include coefficients or powers, apply multiplication carefully:

• $$3(2x^2 + 5x) = 6x^2 + 15x$$
• $$-2(4x^3 - x^2) = -8x^3 + 2x^2$$

Special Products

Square of a Binomial

Formula: $$(a + b)^2 = a^2 + 2ab + b^2$$

Example: $$(x + 5)^2 = x^2 + 10x + 25$$

Difference of Squares

Formula: $$(a + b)(a - b) = a^2 - b^2$$

Example: $$(x + 7)(x - 7) = x^2 - 49$$

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Worked Examples

Example 1 (Foundation): Single bracket

Simplify: $$4(x + 3)$$

• Multiply each term inside the bracket: 4 × x = 4x, 4 × 3 = 12
• Answer: $$4x + 12$$

Example 2 (Foundation): Single bracket with negative

Simplify: $$-2(y - 5)$$

• -2 × y = -2y, -2 × -5 = +10
• Answer: $$-2y + 10$$

Example 3 (Higher): Double brackets

Simplify: $$(x + 2)(x + 3)$$

• x × x = x^2
• x × 3 = 3x
• 2 × x = 2x
• 2 × 3 = 6
• Combine: $$x^2 + 5x + 6$$

Example 4 (Higher): Double brackets with negative

Simplify: $$(x - 4)(x + 6)$$

• x × x = x^2
• x × 6 = 6x
• -4 × x = -4x
• -4 × 6 = -24
• Combine: $$x^2 + 2x - 24$$

Example 5 (Higher): Square of a binomial

Simplify: $$(x + 7)^2$$

• Formula: $$(a + b)^2 = a^2 + 2ab + b^2$$
• $$x^2 + 2 × x × 7 + 49 = x^2 + 14x + 49$$

Example 6 (Higher): Difference of squares

Simplify: $$(x + 5)(x - 5)$$

• $$x^2 - 25$$

Example 7 (Higher): With coefficients and powers

Simplify: $$3(2x^2 + 5x)$$

• 3 × 2x^2 = 6x^2
• 3 × 5x = 15x
• Answer: $$6x^2 + 15x$$

Example 8 (Higher): Negative sign outside bracket

Simplify: $$- (4x^2 - 3x + 5)$$

• - × 4x^2 = -4x^2
• - × -3x = +3x
• - × 5 = -5
• Answer: $$-4x^2 + 3x - 5$$

Example 9 (Real-life application)

A rectangle has length $$(x + 3)$$ and width $$(x + 5)$$. Find the area in expanded form.

• Area = length × width = $$(x + 3)(x + 5)$$
• Expand: $$x^2 + 5x + 3x + 15 = x^2 + 8x + 15$$

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Common Mistakes

• Failing to multiply each term inside the bracket
• Ignoring negative signs outside brackets
• Forgetting to combine like terms after expansion
• Confusing formula for square of a binomial and difference of squares
• Making errors when multiplying coefficients with powers

Tips to avoid errors:

• Always multiply each term individually
• Check signs carefully
• Combine like terms at the end
• Memorize special product formulas
• Practice with numbers and variables together

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Applications

• Algebra: Simplifying expressions to solve equations
• Geometry: Expanding brackets to find areas and perimeters
• Physics: Algebraic formulas involving multiple terms
• Economics/Finance: Expressions for cost, profit, and revenue calculations
• Exams: Required for higher-level algebra manipulation

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Strategies & Tips

• Start by multiplying one term at a time
• Keep track of positive and negative signs
• Combine like terms systematically
• Use FOIL method for double brackets
• Apply special products formulas where appropriate

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Summary / Call-to-Action

Expanding brackets is essential in algebra and real-life applications. By mastering single brackets, double brackets, negative signs, coefficients, powers, and special products, students can simplify expressions effectively. Regular practice ensures accuracy and confidence in solving algebraic problems.

Next Steps:

• Attempt quizzes on expanding brackets to reinforce learning
• Practice single and double bracket expansions
• Use special products to simplify common expressions
• Challenge yourself with mixed expressions including coefficients and negative signs

Consistent practice will make expanding brackets intuitive and error-free in all GCSE Maths problems.