Expanding Brackets

What does expanding mean?

When a number or variable sits directly in front of a bracket, it is multiplying everything inside that bracket. Expanding (also called multiplying out) means doing that multiplication so the brackets disappear.

For example, 3(x + 4) means "3 lots of (x + 4)." Expanding it gives 3x + 12. The two expressions are equal — they just look different.

This builds directly on collecting like terms from Simplifying Expressions, because after expanding you usually tidy up.

The distributive law

The rule behind expanding is the distributive law:

$$ a(b + c) = ab + ac $$

The term outside is "distributed" to each term inside. Picture an arrow going from the outside number to every term within the bracket — and make sure none get left out.

Why does this work? Think of 3(x + 4) as area. A rectangle 3 tall and (x + 4) wide can be split into a 3-by-x piece and a 3-by-4 piece. Their areas, 3x and 12, add to the whole: 3x + 12.

Worked example 1 — a basic expansion

Expand 5(x + 2).

1. Multiply the outside 5 by the first inside term: 5 × x = 5x.
2. Multiply the outside 5 by the second inside term: 5 × 2 = 10.
3. Join with the sign that was inside: 5x + 10.

Worked example 2 — subtraction inside

Expand 4(x − 3).

1. 4 × x = 4x.
2. 4 × (−3) = −12. Keep the minus sign.
3. Result: 4x − 12.

Treat the term as −3, not 3. Carrying the sign along prevents the most common error.

Worked example 3 — a negative outside the bracket

Expand −2(x + 6).

A negative outside flips the sign of everything inside.

1. −2 × x = −2x.
2. −2 × 6 = −12.
3. Result: −2x − 12.

Now try −3(x − 4):

1. −3 × x = −3x.
2. −3 × (−4) = +12 (negative times negative is positive).
3. Result: −3x + 12.

The double-negative becoming positive trips up many students, so slow down whenever you see two minus signs interacting.

Worked example 4 — a variable outside

Expand x(x + 5).

The multiplier can be a variable too.

1. x × x = x² (a variable times itself gives a square).
2. x × 5 = 5x.
3. Result: x² + 5x.

If you'd like a refresher on what x² means, see Exponents and Powers.

Worked example 5 — expand and simplify

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

When there are two brackets added together, expand each one, then collect like terms.

1. Expand the first bracket: 3(x + 2) = 3x + 6.
2. Expand the second bracket: 2(x + 4) = 2x + 8.
3. Write the full expression: 3x + 6 + 2x + 8.
4. Collect like terms: (3x + 2x) + (6 + 8) = 5x + 14.

Final answer: 5x + 14.

Worked example 6 — a minus between two brackets

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

The minus in front of the second bracket acts like multiplying that bracket by −2.

1. First bracket: 5(x + 3) = 5x + 15.
2. Second bracket: −2(x + 1) = −2x − 2.
3. Combine: 5x + 15 − 2x − 2.
4. Collect like terms: (5x − 2x) + (15 − 2) = 3x + 13.

Final answer: 3x + 13.

If you ignore the minus and write +2x + 2, you'll get the wrong answer — so always let the subtraction sign join the second bracket before expanding.

Activity — expand each expression

1. 2(x + 7)
2. 6(x − 1)
3. −3(x + 5)
4. x(x − 2)
5. 4(x + 2) + 3(x + 1)
6. 7(x + 4) − 5(x + 2)

Answers: 1) 2x + 14 2) 6x − 6 3) −3x − 15 4) x² − 2x 5) 7x + 11 6) 2x + 18

Why this matters

Expanding brackets is essential for solving equations with brackets, for working with formulas, and later for quadratics. It is the inverse of factorising — expanding pushes brackets out into a sum, while factorising pulls a sum back into brackets.

Remember three things and you'll rarely go wrong: multiply every inside term, carry the signs (especially a negative outside), and collect like terms at the end. Take the signs slowly and the rest follows naturally.