Expand & Conquer

Expand & Conquer 🗡️

Almost every algebra question rests on two moves that are **reverses** of each other: • **Expand** — remove brackets by multiplying out: 3(x + 4) = 3x + 12. • **Factorise** — put brackets back by taking out common parts: 3x + 12 = 3(x + 4). Get fluent both ways, watch the signs, and most of the algebra paper opens up.

Name the parts 📖

Examiners use these words precisely, so you should too: • **Expression** — terms with no equals sign, e.g. 3x + 2. • **Equation** — has an equals sign and can be solved, e.g. 3x + 2 = 8. • **Identity** — true for *every* value, written with ≡, e.g. 2(x + 1) ≡ 2x + 2. • **Formula** — a rule linking quantities, e.g. A = πr². In a **term** like 5x, the **5** is the **coefficient** of x.

Match the word to the example

Expanding a single bracket 📦

To expand a bracket, multiply **every** term inside by the term outside: **3(x + 4)** = (3 × x) + (3 × 4) = **3x + 12**. The danger is **signs**. A negative outside flips the sign of each term inside:\n\n**−2(x − 3)** = (−2 × x) + (−2 × −3) = **−2x + 6**. (Two negatives make the +6.)

Mind the signs

Expand −2(x − 3).

• −2x + 6
• −2x − 6
• −2x − 3
• 2x − 6

Substitute in

An interactive activity.

Expanding two brackets 🔲

For two brackets, every term in the first must multiply every term in the second — a **grid** keeps it organised. Take **(x + 2)(x + 5)**: • x × x = x² • x × 5 = 5x • 2 × x = 2x • 2 × 5 = 10 Add them and collect like terms: x² + 5x + 2x + 10 = **x² + 7x + 10**.

Order the expansion

An interactive activity.

The squaring trap

Expand (x + 3)². Remember it means (x + 3)(x + 3).

• x² + 6x + 9
• x² + 9
• x² + 3x + 9
• x² + 6x + 6

Difference of two squares ➗

One product is worth memorising because the middle terms **cancel**: **(x + 4)(x − 4)** = x² − 4x + 4x − 16 = **x² − 16**. Whenever you see (a + b)(a − b), the answer is just **a² − b²** — the *difference of two squares*. You will use it constantly, forwards and backwards.

Expanding rules

To expand a single bracket, multiply _____ term inside by the term outside. When you square a bracket such as (x + 3)², you must keep the _____ term, giving x² + 6x + 9. And (x + 4)(x − 4) gives the difference of two _____: x² − 16.

Factorising: expanding in reverse 🏭

**Factorising** undoes expanding — put the brackets back. Always look for a **common factor** first: **6x + 9**: both terms divide by 3, so 6x + 9 = **3(2x + 3)**. Check by expanding: 3 × 2x = 6x and 3 × 3 = 9. Take out the **highest** common factor, or you will not be fully factorised.

Factorise fully

Factorise 6x + 9 completely.

• 3(2x + 3)
• 3(x + 3)
• 6(x + 3)
• 3(2x + 9)

Factorising x² + bx + c 🧩

To factorise a quadratic like **x² + 7x + 10**, find two numbers that **multiply to c** and **add to b**: • multiply to 10 and add to 7 → **2 and 5**. So x² + 7x + 10 = **(x + 2)(x + 5)**. Check by expanding: the cross terms 5x + 2x give 7x. This is the reverse of the two-bracket expansion you just practised.

Match each quadratic to its factorised form