🍬 Expanding Brackets

Cambridge Lower Secondary · Grade 7 · Algebra

Single Bracket
3(2x + 5) = 6x + 15
Multiply every term inside by the factor outside
Negative Outside
−2(x + 3) = −2x − 6
Negative × positive = negative
Factorise (Reverse)
6x + 9 = 3(2x + 3)
Find the HCF, then divide each term

Watch the distribution! Click to animate 🍭

3 ( 2x + 4 )

What you'll learn:

  • Expanding a single bracket: a(b + c) = ab + ac
  • Expanding when the outside term is negative: −2(x + 3) = −2x − 6
  • Expanding and simplifying two brackets: 3(2x + 1) + 2(x − 4)
  • Expanding with fractional and decimal coefficients
  • Factorising by finding the HCF: 6x + 9 = 3(2x + 3)
  • The area model — connecting rectangles to algebra
Created by Mr Josh

📖 Learn: Expanding Brackets

Part 1: The Distributive Law

To expand a bracket means to multiply the term outside by every term inside.

📌 Rule: a(b + c) = ab + ac
Example:   3(x + 4)  →  3 × x + 3 × 4  =  3x + 12
Example:   5(2y − 3)  →  5 × 2y + 5 × (−3)  =  10y − 15
Example:   4(3a + 2b)  →  12a + 8b
💡 Think of it like sharing a bag of sweets — the number outside shares with everyone inside the bracket.

Part 2: Negative Outside the Bracket

When the outside term is negative, take extra care with signs.

📌 Remember: negative × positive = negative
📌 Remember: negative × negative = positive
Example:   −2(x + 3)  →  −2x − 6
Example:   −3(4 − y)  →  −12 + 3y  =  3y − 12
Example:   −(5x − 2)  →  −5x + 2
💡 A minus sign in front of a bracket flips the sign of every term inside: −(a − b) = −a + b

Part 3: Expanding and Simplifying

Expand each bracket, then collect like terms.

Example:   3(2x + 1) + 2(x − 4)
Step 1 — Expand:   6x + 3 + 2x − 8
Step 2 — Collect x-terms:   6x + 2x = 8x
Step 3 — Collect numbers:   3 − 8 = −5
Answer:   8x − 5
💡 Always expand first, then simplify. Never try to do both at once!

Part 4: Fractional and Decimal Coefficients

The same rule applies — multiply the outside term by every term inside.

Example:   ½(6x + 4)  →  3x + 2
Example:   0.5(8x − 6)  →  4x − 3
Example:   ⅓(9a + 12b)  →  3a + 4b
Example:   1.2(5x + 10)  →  6x + 12
💡 Fraction × number: simplify by dividing. e.g. ½ × 6x = 3x because 6 ÷ 2 = 3

Part 5: Factorising — The Reverse

Factorising is the reverse of expanding. Find the HCF (Highest Common Factor) of all terms, then write what remains inside the bracket.

Expression HCF Factorised Form
6x + 9 3 3(2x + 3)
8a − 12 4 4(2a − 3)
15x + 10y 5 5(3x + 2y)
x² + 3x x x(x + 3)
💡 Check: always expand your answer back out — you should get the original expression!
Created by Mr Josh

💡 Worked Examples

Example 1: Expanding a Single Bracket

Expand: 4(3x − 2)

Step 1: Multiply 4 by 3x  →  4 × 3x = 12x
Step 2: Multiply 4 by −2  →  4 × (−2) = −8
Answer: 4(3x − 2) = 12x − 8
📌 Every term inside the bracket gets multiplied by the term outside.

Example 2: Negative Outside the Bracket

Expand: −3(2x − 5)

Step 1: Multiply −3 by 2x  →  −3 × 2x = −6x
Step 2: Multiply −3 by −5  →  −3 × (−5) = +15
Answer: −3(2x − 5) = −6x + 15
📌 Negative × negative = positive. The sign flip is crucial!

Example 3: Expanding and Simplifying

Expand and simplify: 5(x + 2) − 3(x − 1)

Step 1: Expand first bracket:   5x + 10
Step 2: Expand second bracket:   −3x + 3
Step 3: Write together:   5x + 10 − 3x + 3
Step 4: Collect x-terms:   5x − 3x = 2x
Step 5: Collect numbers:   10 + 3 = 13
Answer: 2x + 13

Example 4: Factorising Fully

Factorise fully: 12x + 18

Step 1: Find factors of 12: 1, 2, 3, 4, 6, 12
Step 2: Find factors of 18: 1, 2, 3, 6, 9, 18
Step 3: HCF = 6
Step 4: Divide each term by 6:   12x ÷ 6 = 2x  and  18 ÷ 6 = 3
Answer: 12x + 18 = 6(2x + 3)
✅ Check: 6 × 2x + 6 × 3 = 12x + 18 ✔
Created by Mr Josh

🎨 Interactive Visualizer

Arrow Animator: Watch Distribution Happen

Enter values for a(bx + c) and watch animated arrows fly from the outside term to each term inside.

Area Model: Rectangle Algebra

The area of the whole rectangle = a(b + c). The two smaller rectangles have areas ab and ac.

Try these in the arrow animator:

Created by Mr Josh

✏️ Exercise 1: Basic Expanding

Expand the bracket. Write your answer in the form ax + b or ax + by with no spaces (e.g. 6x+12 or 6x-12).

Created by Mr Josh

✏️ Exercise 2: Expanding with Negative Factors

Take special care with signs. Remember: negative × negative = positive.

Created by Mr Josh

✏️ Exercise 3: Expand and Simplify

Expand both brackets, then collect like terms. Write your simplified answer.

Created by Mr Josh

🧩 Exercise 4: Factorising — Find the HCF!

Find the HCF of the expression, then complete the bracket. Both boxes must be correct.

Created by Mr Josh

🔀 Exercise 5: Mixed — Expand, Simplify & Factorise

A mix of all skills. Read each question carefully!

Created by Mr Josh

📝 Practice Questions

  1. Expand: 3(x + 5)
  2. Expand: 4(2x − 3)
  3. Expand: 7(3a + 2b)
  4. Expand: 5(4 − x)
  5. Expand: −2(x + 6)
  6. Expand: −3(2x − 4)
  7. Expand: −(5x + 1)
  8. Expand: −4(3 − 2y)
  9. Expand and simplify: 2(3x + 1) + 4(x − 2)
  10. Expand and simplify: 5(x + 3) − 2(x + 1)
  11. Expand and simplify: 3(2x − 1) − (x + 4)
  12. Expand: ½(8x + 6)
  13. Expand: 0.4(10x − 5)
  14. Expand: ⅓(12a + 9b)
  15. Factorise: 4x + 8
  16. Factorise: 6a − 9
  17. Factorise: 10x + 15y
  18. Factorise: 3x² + 12x
  19. Factorise fully: 16a + 24b
  20. A rectangle has length (2x + 5) and width 3. Write its area as an expanded expression.
  1. 3x + 15
  2. 8x − 12
  3. 21a + 14b
  4. 20 − 5x
  5. −2x − 12
  6. −6x + 12
  7. −5x − 1
  8. −12 + 8y  (or 8y − 12)
  9. 6x + 2 + 4x − 8 = 10x − 6
  10. 5x + 15 − 2x − 2 = 3x + 13
  11. 6x − 3 − x − 4 = 5x − 7
  12. 4x + 3
  13. 4x − 2
  14. 4a + 3b
  15. 4(x + 2)
  16. 3(2a − 3)
  17. 5(2x + 3y)
  18. 3x(x + 4)
  19. 8(2a + 3b)
  20. 3(2x + 5) = 6x + 15
Created by Mr Josh

🏆 Challenge Questions

  1. Expand and simplify: 4(2x + 3) − 2(3x − 1) + 5x
  2. Expand and simplify: 3(a + 2b) − 2(3a − b)
  3. A rectangle has sides (3x + 4) and 2, and a square has side length x. Write an expression for the total area, fully simplified.
  4. Factorise fully: 24x² + 36x
  5. Show that 2(3x + 5) − (2x + 3) ≡ 4x + 7 for all values of x.
  6. If 5(x + k) = 5x + 20, find the value of k.
  7. The perimeter of a triangle is (12x + 18). One side is 4x + 6. Another side is 2x + 3. Write an expression for the third side, factorised.
  8. Expand and fully simplify: ½(4x − 6) + ⅓(9x + 12) − 2(x − 1)
  1. 8x + 12 − 6x + 2 + 5x = 7x + 14
  2. 3a + 6b − 6a + 2b = −3a + 8b
  3. Rectangle area = 2(3x + 4) = 6x + 8; Square area = x²; Total = x² + 6x + 8
  4. HCF = 12x; 12x(2x + 3)
  5. LHS = 6x + 10 − 2x − 3 = 4x + 7 = RHS ✔
  6. 5 × k = 20, so k = 4
  7. Third side = 12x + 18 − (4x + 6) − (2x + 3) = 6x + 9 = 3(2x + 3)
  8. 2x − 3 + 3x + 4 − 2x + 2 = 3x + 3
Created by Mr Josh