Expand & Conquer

Expand & Conquer

Almost every algebra question rests on two moves that are reverses of each other: expand removes brackets by multiplying out, 3(x + 4) = 3x + 12, and factorise puts them back, 3x + 12 = 3(x + 4). Get fluent both ways, watch the signs, and most of the algebra paper opens up. Examiners use these words precisely, so you should too:

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

Substitute x = 5 into the expression 3(x + 4). What value does it give?

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

Put the steps of expanding (x + 2)(x + 5) in order.

• Multiply the first terms: x × x = x²
• Multiply the outer terms: x × 5 = 5x
• Multiply the inner terms: 2 × x = 2x
• Multiply the last terms: 2 × 5 = 10
• Collect like terms: x² + 5x + 2x + 10 = x² + 7x + 10

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, and take out the highest one 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.

Pick your weapon

Three expressions, no instructions. For each one, choose the move you would make first.

• Factorise fully: 12x + 18
• Factorise: x² − 49
• Factorise: x² + 9x + 20

Match each quadratic to its factorised form