Expanding a Single Bracket

Suppose a school orders 3 lunch packs, and every pack holds a sandwich costing x pounds plus a fixed £4 drink-and-fruit box. How much is the whole order? You could work out one pack and multiply by three — or you could notice the total is 3 lots of (x + 4), written 3(x + 4).

Expanding that bracket means multiplying everything inside by the term outside, so 3(x + 4) = 3x + 12: three sandwiches plus £12 of boxes. This one move — the distributive law — is among the most-used steps in all of algebra, and it rests on the same idea as finding the area of a rectangle that has been split into two.

a(b + c) = ab + ac

It is the same distributive law you already use for numbers — splitting one multiplication into two easy ones — only now one of the terms is an unknown.

Worked example 1 — a number outside

Take 3(x + 4). Multiply the outside 3 by each term inside — by the x, then by the 4:

3(x + 4) = 3 \times x + 3 \times 4 = 3x + 12

We cannot add 3x and 12 together — they are not like terms — so 3x + 12 is the finished, expanded form.

See it as an area

A rectangle 3 tall and x + 4 wide has area 3(x + 4). Slice it along the line between the x part and the 4 part and you get two rectangles: one 3 \times x = 3x and one 3 \times 4 = 12. Nothing moved, so the two areas must add up to the whole. Step through it.

Worked example 2 — a letter outside

The same trick works when the outside term is itself a letter. Multiply the x by each term inside:

x(x + 5) = x \times x + x \times 5 = x^2 + 5x

Here x \times x = x^2 (an x times an x), and x \times 5 = 5x. So multiplying a bracket by a letter can raise the power — a plain x outside turns the inside x into an x^2.

Worked example 3 — mind the minus

When the outside term is negative, that minus sign multiplies every term inside too. Expand -2(x - 3):

-2(x - 3) = (-2) \times x + (-2) \times (-3) = -2x + 6

The first term is -2 \times x = -2x. The second is -2 \times (-3), and a minus times a minus is a plus, so it becomes +6. The finished form is -2x + 6. Signs are where most slips happen — go slowly and track each one.

Worked example 4 — expand, then collect

Often you expand two brackets and then tidy up. Take 2(x + 3) + 3(x + 1). Expand each bracket first:

2(x + 3) = 2x + 6, \qquad 3(x + 1) = 3x + 3

Now add them and collect like terms — the x's together, the plain numbers together:

2x + 6 + 3x + 3 = 5x + 9

Expand first, tidy second — a two-step move you will use constantly.

These two mistakes trip up almost everyone learning to expand:

Multiply the outside by EVERY term inside — not just the first one. A very common error is 3(x + 4) = 3x + 4. That leaves the 4 untouched, but the 3 multiplies it too: the correct answer is 3x + 12. Draw a little arrow from the outside term to each inside term so none gets forgotten.
Take great care with signs. With -2(x - 3) the minus outside hits both terms. The second product is -2 \times -3 = +6 (minus times minus is plus), giving -2x + 6 — not -2x - 6. This is exactly where careful students still slip, so pause on the second term every time.

You already do this trick with numbers, probably without noticing. To work out 7 \times 102 in your head, you split the 102 into 100 + 2 and multiply each part:

7 \times 102 = 7 \times (100 + 2) = 700 + 14 = 714

That is exactly expanding a bracket — the distributive law a(b + c) = ab + ac — with real numbers instead of letters. Algebra just gives the same familiar move a name and lets one of the pieces be an unknown.

Expanding takes a bracket apart: 3(x + 4) \to 3x + 12. Run the arrow the other way — spot what 3x and 12 share and pull it back out to the front — and you get 3(x + 4) again. That reverse move is called factorising: expanding pulls brackets apart, factorising puts them back together.

Being fluent in both directions is what unlocks the next big room of algebra — quadratics, algebraic fractions, and solving equations all lean on switching freely between a bracketed form and an expanded one.