How to Solve: Expanding Brackets

How to Solve: Expanding Brackets

(For Students Who Want to Ace Their Exam & Teachers Who Need a Ready-to-Record Script)

Introduction

"If you can’t expand brackets correctly, you’ll lose marks on every algebra question—from simplifying expressions to solving equations. Let’s fix that in 5 minutes."

What You Need To Know First

Before expanding brackets, you must already understand: 1. Order of operations (BIDMAS/BODMAS) – Multiplication before addition/subtraction. 2. Multiplying terms – How to multiply numbers, variables (e.g., x × x = x²), and coefficients (e.g., 3 × 4x = 12x). 3. Like terms – Terms with the same variable part (e.g., 2x and 5x are like terms; 2x and 3y are not).

If any of these are unclear, review them first.

Key Vocabulary

Formulas To Know

1. Single Bracket Expansion

Formula: a(b + c) = ab + ac - a = term outside the bracket (can be a number, variable, or both). - b, c = terms inside the bracket.

MEMORISE THIS – It’s the foundation of all bracket expansion.

2. Double Bracket Expansion (FOIL Method)

Formula: (a + b)(c + d) = ac + ad + bc + bd - a, b = terms in the first bracket. - c, d = terms in the second bracket.

MEMORISE THIS – Examiners test this frequently.

3. Squaring a Bracket (Special Case)

Formula: (a + b)² = a² + 2ab + b² (a – b)² = a² – 2ab + b²

MEMORISE THIS – Saves time in exams.

4. Difference of Two Squares

Formula: (a + b)(a – b) = a² – b²

MEMORISE THIS – A shortcut for specific cases.

Step-by-Step Method

How to Expand Any Bracket

Follow these steps in order for every problem:

1. Identify the type of bracket – Is it a single bracket (3(x + 2)) or double bracket ((x + 1)(x + 3))?
2. Apply the correct formula – Use the single bracket rule or FOIL for double brackets.
3. Multiply each term inside by the term outside (for single brackets) or multiply each term in the first bracket by each term in the second (for double brackets).
4. Write all new terms – Don’t skip any!
5. Combine like terms – Add or subtract terms with the same variable.
6. Simplify the final expression – Write in standard form (highest power first).

Worked Example (Using the Steps)

Problem: Expand 3(2x – 5) + 4(x + 1)

Step 1: Identify the type of bracket. - Two single brackets: 3(2x – 5) and 4(x + 1).

Step 2: Apply the single bracket formula to each. - 3 × 2x = 6x - 3 × (-5) = -15 - 4 × x = 4x - 4 × 1 = 4

Step 3: Write all new terms. - 6x – 15 + 4x + 4

Step 4: Combine like terms. - 6x + 4x = 10x - -15 + 4 = -11

Step 5: Simplify. - 10x – 11

Final Answer: 10x – 11

Worked Examples

Example 1 – Basic (Single Bracket)

Problem: Expand 5(3x – 2)

Working: 1. Multiply 5 by 3x → 15x 2. Multiply 5 by -2 → -10 3. Combine: 15x – 10

What we did and why: - Used the single bracket rule (a(b + c) = ab + ac). - Multiplied the term outside (5) by each term inside (3x and -2). - No like terms to combine, so the answer is already simplified.

Example 2 – Medium (Double Bracket)

Problem: Expand (x + 4)(x – 2)

Working: 1. First terms: x × x = x² 2. Outer terms: x × (-2) = -2x 3. Inner terms: 4 × x = 4x 4. Last terms: 4 × (-2) = -8 5. Combine all: x² – 2x + 4x – 8 6. Combine like terms: x² + 2x – 8

What we did and why: - Used the FOIL method (First, Outer, Inner, Last). - Multiplied every term in the first bracket by every term in the second bracket. - Combined -2x and +4x to get +2x.

Example 3 – Exam Style (Disguised Problem)

Problem: Simplify 2(3x + 1) – (x – 4)

Working: 1. Expand 2(3x + 1) → 6x + 2 2. Expand –(x – 4) → -1 × x = -x and -1 × (-4) = +4
(Note: The minus sign is like a -1 outside the bracket!) 3. Combine all terms: 6x + 2 – x + 4 4. Combine like terms: 6x – x = 5x and 2 + 4 = 6 5. Final answer: 5x + 6

What we did and why: - Treated the minus sign as -1 to avoid sign errors. - Expanded both brackets separately before combining. - Combined like terms carefully to simplify.

Common Mistakes

Exam Traps

1-Minute Recap

"Okay, let’s lock this in. Expanding brackets is just multiplying—nothing more. For single brackets like 3(x + 2), multiply 3 by both x and 2 to get 3x + 6. For double brackets like (x + 1)(x + 3), use FOIL: First, Outer, Inner, Last. Multiply x × x, x × 3, 1 × x, and 1 × 3, then combine like terms. Watch out for minus signs—they’re sneaky! Always rewrite - (x + 2) as -1(x + 2) before expanding. And if you see a squared bracket like (x + 4)², use the shortcut: x² + 8x + 16. Practice 3 problems tonight, and you’ll own this for the exam. You’ve got this!