KS3 · Maths

Expanding a single bracket

3 × 12 is 36, obviously. So why do so many people write 3(y + 2) = 3y + 2, which says it's 32? Let's make sure you never do.

Algebra · Expanding brackets

Equivalent or broken?

Does each expansion still equal the bracket it came from? Stuck? Put in a number, like 10, and work out both sides.

Still to sort

Equivalent: every term was multiplied (0)

Both sides give the same answer, whatever number you put in.

Where the line is: Swapping the order of an ADDITION is fine. Swapping the order of a SUBTRACTION is not.

Not equivalent: something went wrong (0)

A term escaped being multiplied, or a rule was bent.

Where the line is: It only takes one term that was not multiplied to break the whole expansion.

8 of 8 still to sort.

Eight students each expanded a bracket. Some kept the value exactly the same. Some quietly broke it. Sort them, then read why.

Build it yourself

From numbers to letters

You already do this with numbers. 3 × 57 is 3 lots of 50 plus 3 lots of 7: 150 + 21 = 171. Swap a number for a letter and the method stays the same. Fill in the grid for 12(y + 7).

Grid: each cell is its row times its column
y7
12

Type x² as x^2 if you cannot type ². Spaces do not matter.

Level up

When the multiplier is a letter term

Now the term outside is 2x and there are three terms inside. Same rule: 2x multiplies every column. Take the minus sign with the 4y, and think hard about the last cell: what is x × x?

Grid: each cell is its row times its column
3−4y7x
2x

Type x² as x^2 if you cannot type ². Spaces do not matter.

Spot the slip

Where does the minus sign go?

Expand and simplify 5(y + 1) − 2(y + 3).

Jamie's answer — which line goes wrong?

Your turn to finish it

Two brackets, one tidy answer

Expand and simplify 2(x + 4) + 5(x + 3).

  1. Expand the first bracket: 2(x + 4) = 2x + 82 lots of x and 2 lots of 4.
  2. missing step
Which line is step 2?

Predict, then check

Before you expand anything, look closely: both brackets are exactly the same.

3(x + 5) + 4(x + 5) is 3 lots of (x + 5) plus 4 lots of (x + 5). What does it simplify to?

WHAT YOU'VE LEARNED

A quick recap of today's lesson.

The term outside multiplies everything inside. Nobody gets left out.

What you need to know

  • Expanding (or multiplying out) a bracket means writing an equivalent expression without the bracket.
  • The term outside multiplies EVERY term inside: 3(x + 5) = 3x + 15, and 4(2a − 3b + 5) = 8a − 12b + 20.
  • x × x = x², not 2x. Powers multiply the same way: x²(x + 1) = x³ + x², because x² × x = x × x × x.
  • A negative multiplier changes the signs: −2(x + 5) = −2x − 10, and −5(x − 7) = −5x + 35, because a negative times a negative is positive.
  • With more than one bracket, expand each bracket, then collect like terms: 2(x + 5) + 3(x + 6) = 2x + 10 + 3x + 18 = 5x + 28.

The big picture

Expanding a bracket means multiplying the term outside by every term inside, so the new expression is equal to the old one. Constants, letters, negatives and powers all follow the same rule: x × x = x², a negative multiplier changes the signs, and with two or more brackets you expand each one and then collect like terms.

Key points

1A constant is a term that never changes, like the 5 in x + 5. A variable, like x, can take a range of values. Inside a bracket, both get multiplied.
2Order does not matter for addition or multiplication, so 3(2 + a) and (a + 2) × 3 both give 3a + 6. It does matter for subtraction: 3(a − 2) = 3a − 6, but 3(2 − a) = 6 − 3a.
3Brackets change the meaning. x² × x + 7 = x³ + 7, but x²(x + 7) = x³ + 7x², because the bracket makes x² multiply the 7 as well.
4When a bracket is subtracted, its minus sign multiplies every term inside: 6(x + 5) − 3(x + 2) = 6x + 30 − 3x − 6 = 3x + 24.
5Matching brackets can be combined before you expand: 5(x + 1) + 4(x + 1) = 9(x + 1) = 9x + 9.

Worked example

Problem

Expand 3y(x − 7 + 5y).

⚠ Watch out

Multiplying only the first term in the bracket, so 3(y + 2) becomes 3y + 2 instead of 3y + 6. The number outside multiplies everything inside, and a quick number test catches it: with y = 10, 3 × 12 is 36, not 32.

🧠

Memory hook

The term outside the bracket shakes hands with every term inside. Nobody gets left out, and the handshake is a multiply. If you are not sure, put in 10: if the two sides give different answers, the expansion is broken.

✓

Check yourself

Cover the page. Expand a(a − 3) and 4(a + b + c + 3), then test one answer with a = 10. (Answers: a² − 3a and 4a + 4b + 4c + 12.)

Flashcards

(14)
What does it mean to expand (or multiply out) a bracket?
Write an equivalent expression without the bracket, one that is equal to the original for every value.
In x + 5, which term is a constant and which is a variable?
5 is the constant: it never changes. x is the variable: it can take a range of values.
How does the distributive law make 4 × 39 easier?
4 lots of 30 plus 4 lots of 9: 120 + 36 = 156.
Why is 3(y + 2) not 3y + 2?
You need 3 lots of EVERYTHING in the bracket: 3 lots of y and 3 lots of 2. So 3(y + 2) = 3y + 6.
What is x × x?
x², not 2x. (2x means x + x.)
How do 3(100 + 2) and 3(100 − 2) show what a subtraction inside a bracket does?
3(100 + 2) = 300 + 6 = 306, but 3(100 − 2) = 300 − 6 = 294. When the bracket holds a subtraction, the product is subtracted.
Do 4(x + y) and (y + x) × 4 mean the same thing?
Yes. Addition and multiplication work in any order, so both are 4x + 4y.
Is 3(2 − a) the same as 3(a − 2)?
No. Subtraction does not work in any order: 3(2 − a) = 6 − 3a, but 3(a − 2) = 3a − 6.
Expand −9(x − 3).
−9x + 27. −9 × x = −9x, and −9 × −3 = +27 because a negative times a negative is positive.
Expand y²(y + 4).
y³ + 4y². y² × y = y × y × y = y³, and y² × 4 = 4y².
What are like terms?
Terms with the same letters raised to the same powers, like 3x and 7x. x² and x are NOT like terms, so x² + 5x cannot be simplified.
What is the method for 6(x + 1) + 7(x + 3)?
Expand each bracket, then collect like terms: 6x + 6 + 7x + 21 = 13x + 27.
Simplify 5(x − 7) + 2(x − 7) before expanding.
5 lots plus 2 lots of the same bracket is 7 lots: 7(x − 7) = 7x − 49.
Simplify 10(x + 2) + 10(x + 4) using the matching multiplier.
10(x + 2 + x + 4) = 10(2x + 6) = 20x + 60.

Tap any card to flip it, or use Study as deck to go through them one at a time. In the full lesson these run as a spaced-repetition deck — you rate each card Hard, Good or Easy and the tricky ones keep coming back until they stick.

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