Module 7 of 30Lesson 1 of 35 · A-C1 Algebraic Manipulation I
Algebra strand · Unit A-C1 · Syllabus C2.2

Algebraic Manipulation I

🔵 Core levelLetters outside the brackets

1 Learn it

Core manipulation combines all your Foundation skills at exam speed. The step up: the multiplier outside a bracket can now be a LETTER or a whole term.

\(x(x + 5) = x^{2} + 5x\)  ·  \(3x(2x – 4y) = 6x^{2} – 12xy\)

🔑 Deliver the WHOLE term: in \(3x(2x – 4y)\), the 3x visits both terms — numbers multiply (3×2=6, 3×4=12), letters multiply (x·x = x², x·y = xy). Track number, letter and SIGN separately on each delivery.

The syllabus’s own benchmark: simplify \(2a + 3b + 5a – 9b = 7a – 6b\). Signs travel with their terms — the 9 keeps its minus.

2 Watch it work
EXAMPLE 1

Simplify \(2a + 3b + 5a – 9b\).

a-family: 7a · b-family: 3b − 9b = −6b → \(7a – 6b\)

EXAMPLE 2

Expand \(x(x + 5)\).

\(x \cdot x + x \cdot 5 = x^{2} + 5x\)

EXAMPLE 3

Expand \(3x(2x – 4y)\).

\(3x \cdot 2x = 6x^{2}\) and \(3x \cdot (-4y) = -12xy\) → \(6x^{2} – 12xy\)

EXAMPLE 4

Expand and simplify \(2x(x + 3) – x(x – 1)\).

\(2x^{2} + 6x – x^{2} + x\) — note the double flip: \(-x \times (-1) = +x\).

\(= x^{2} + 7x\)

3 Your turn — try first, then reveal
🚫 No calculator[1 mark]

Q1. Simplify \(4a + 7b – a + 2b\).

👀 Show answer
✍️ How your working should look in the exam
4a − a = 3a ,  7b + 2b = 9b → 3a + 9b   ✓ B1
Why this works
Two families collected separately; the lone a is 1a.

🚫 No calculator[1 mark]

Q2. Expand \(y(y – 4)\).

👀 Show answer
✍️ How your working should look in the exam
y(y − 4) = y2 − 4y   ✓ B1
Why this works
y · y = y² and y · (−4) = −4y.

🚫 No calculator[1 mark]

Q3. Expand \(2p(3p + 5)\).

👀 Show answer
✍️ How your working should look in the exam
2p(3p + 5) = 6p2 + 10p   ✓ B1
Why this works
2×3 = 6 with p·p = p²; then 2×5 = 10 with the p.

🚫 No calculator[2 marks]

Q4. Expand \(5m(2m – 3n)\).

👀 Show answer
✍️ How your working should look in the exam
5m · 2m = 10m2   ✓ B1
5m · (−3n) = −15mn → 10m² − 15mn   ✓ B1
Why this works
Number × number, letter × letter, sign carried — three checks per delivery.

🚫 No calculator[3 marks]

Q5. Expand and simplify \(3(2x + 1) + 2x(x – 4)\).

👀 Show answer
✍️ How your working should look in the exam
6x + 3 + 2x2 − 8x   ✓ M1 M1
= 2x2 − 2x + 3   ✓ A1
Why this works
Expand both brackets, then collect: 6x − 8x = −2x. Write answers in falling powers: x² term, x term, number.

🚫 No calculator[2 marks]

Q6. Expand and simplify \(x(x + 2) – 3(x + 2)\).

👀 Show answer
✍️ How your working should look in the exam
x2 + 2x − 3x − 6   ✓ M1
= x2 − x − 6   ✓ A1
Why this works
Two deliveries, then 2x − 3x = −x. (Secret preview: this is exactly what expanding the double bracket (x + 2)(x − 3) gives — next unit!)

4 Common mistakes — don’t lose easy marks
  • \(x \cdot x = 2x\): no — multiplying gives \(x^{2}\); adding gives 2x.
  • Dropping the sign on the second delivery: \(3x(2x − 4y)\) ends in −12xy, not +12xy.
  • Only the number multiplies: in 2p(3p + 5) the p must reach both terms: 6p² + 10p, never 6p + 10p.
  • Collecting x² with x: they are different families, always.
5 How the exam asks this

“Expand and simplify” is a guaranteed Core question worth 2–3 marks, and the same manipulation hides inside equations, areas and perimeters everywhere. Next: double brackets — the most-practised expansion in IGCSE history.

✅ Take the Check Yourself quiz below, then continue to A-C2 Double Brackets.

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