Expanding Single Brackets

Grade 3

What expanding means

Expanding a bracket means removing it by multiplying the term outside by every term inside. Think of it as the distributive law in action.

3(x + 4) means 3 lots of (x + 4). So: 3 × x + 3 × 4 = 3x + 12. Simple as that.

Negative terms outside the bracket

When a negative term is outside the bracket, it flips the sign of every term inside. −2(x − 5) = −2x + 10, not −2x − 10. The minus outside hits the minus inside to give a plus.

Worked example

EXAMPLE

Expand and simplify: 3(2x − 4) + 2(x + 5)

Expand first bracket: 6x − 12. Expand second bracket: 2x + 10. Collect like terms: 6x + 2x = 8x, and −12 + 10 = −2. Answer: 8x − 2.

Factorising — the reverse journey

Factorising is the opposite of expanding. You look for the highest common factor of all the terms and pull it outside a bracket.

To factorise 6x + 10: the HCF of 6 and 10 is 2, so the answer is 2(3x + 5). Always check by expanding back — if you get what you started with, you are done.

Try this one

(a) Factorise \( x^2 - 9x + 18 \). [2]

(b) Hence solve \( x^2 - 9x + 18 = 0 \). [1]

Nova's hint:

Part (a) uses factorising a quadratic with a positive constant and a negative middle coefficient.

Ask yourself: 'Which two negative numbers multiply to give \( +18 \) and add to give \( -9 \)?'

Once you have factorised in part (a), use the fact that if a product of two brackets equals zero, then at least one bracket must equal zero.

Try to Solve it with Nova?