Expanding double brackets

Expanding double brackets means multiplying out two brackets, such as (x + 2)(x + 6), so there are no brackets left. Each term of one bracket multiplies each term of the other, then collect like terms: (x + 2)(x + 6) = x² + 6x + 2x + 12 = x² + 8x + 12. Two brackets with two terms each always give four products before you simplify.

How to expand double brackets

1. Take each term of the first bracket and multiply it by both terms of the second. With two terms in each, that is four multiplications.
2. Write all four results down, keeping each sign with its term.
3. Collect like terms. Usually the two x terms combine.

AQA's GCSE Maths specification includes expanding products of two binomials (brackets with two terms each) at both tiers; expanding three or more is Higher only. It matters because it is the reverse of factorising quadratics, and you need it to solve equations and prove identities.

The FOIL method

FOIL is a way to remember the four multiplications in order:

Add them: x² + 6x + 2x + 12 = x² + 8x + 12. Some people draw the four lines linking the terms, which looks like a smiley face or a claw. It is the same four multiplications.

The grid method

Put one bracket across the top of a grid and the other down the side, then multiply to fill each box. It's hard to miss a term this way, and it handles negative numbers well.

Add the four boxes: x² + 6x + 2x + 12 = x² + 8x + 12. The grid is really an area. A rectangle that is x + 6 long and x + 2 wide splits into four smaller rectangles, and the total area is the expansion:

Check it with x = 4: (4 + 2)(4 + 6) = 6 × 10 = 60, and 4² + 8 × 4 + 12 = 16 + 32 + 12 = 60. Putting a number in like this is a quick check on any expansion.

Worked examples

Example 1: with a negative

Expand (x − 7)(x + 3).

• x × x = x²
• x × 3 = 3x
• −7 × x = −7x
• −7 × 3 = −21

Collect: x² + 3x − 7x − 21 = x² − 4x − 21.

Example 2: two negatives

Expand (x − 5)(x − 4). The last term is −5 × −4 = +20, because a negative times a negative is positive. So x² − 4x − 5x + 20 = x² − 9x + 20.

Example 3: a number in front of x

Expand (2x + 5)(x − 1). First 2x × x = 2x², outer 2x × −1 = −2x, inner 5 × x = 5x, last 5 × −1 = −5. Collect: 2x² + 3x − 5.

Example 4: two coefficients

Expand (3x − 4)(2x + 3). The four products are 6x², 9x, −8x and −12, so the answer is 6x² + x − 12. The x terms nearly cancel, which is easy to get wrong if you rush: 9x − 8x = 1x, written just x.

Example 5: two letters

Expand (a + 2b)(3a − b). The products are 3a², −ab, 6ab and −2b². Since ab and ba are the same, −ab + 6ab = 5ab, giving 3a² + 5ab − 2b².

Try one yourself, then press Mark to check it in the Algebra course.

Expanding squared brackets

A squared bracket is a bracket times itself, so write it out twice first:

The pattern is always the same: square the first term, double the product of the two terms, square the last term.

Difference of two squares

When the brackets are the same except for the sign in the middle, the x terms cancel:

In general, (a + b)(a − b) = a² − b². Spotting this the other way round is a quick way to factorise: x² − 49 = (x + 7)(x − 7).

Expanding triple brackets (Higher)

Expand two of the brackets first, then multiply the result by the third. For (x + 1)(x − 2)(x + 4):

1. First two: (x + 1)(x − 2) = x² − 2x + x − 2 = x² − x − 2.
2. Multiply each of those three terms by x, then by 4: x³ − x² − 2x + 4x² − 4x − 8.
3. Collect like terms: x³ + 3x² − 6x − 8.

Three terms times two terms gives six products. Check with x = 1: (2)(−1)(5) = −10, and 1 + 3 − 6 − 8 = −10.

Common mistakes

• Missing the middle terms. (x + 4)(x + 5) has four products, not two: it is x² + 9x + 20, not x² + 20.
• Losing a negative. Keep each sign with the term after it. In (x − 3)(x + 6), the inner product is −3 × x = −3x.
• Squaring wrongly. (x − 6)² ≠ x² − 36 and ≠ x² + 36. Write the bracket out twice.
• Wrong signs on the last term. Negative times negative is positive: −5 × −4 = +20.
• Writing 2x² as (2x)². In (2x + 1)(x + 3), the first product is 2x × x = 2x², not 4x².

The factoring calculator goes the other way, turning a quadratic back into brackets with working, so you can use it to check an expansion. For practice in a short timed paper, try today's GCSE Maths Foundation or GCSE Maths Higher paper.

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