Expanding Double Brackets (Cambridge (CIE) IGCSE Maths)

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Written by: Mark Curtis

Reviewed by: Dan Finlay

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Expanding Two Brackets

How do I expand two brackets using FOIL?

Every term in the first bracket must be multiplied by every term in the second bracket
- Expanding (x + 1)(x + 3) requires 4 multiplications in total
A good way to remember all the multiplications is FOIL
- F = First: multiply together the first terms in each bracket
- O = Outside: multiply the first term in the first bracket by the last term in the last bracket
  - Visually, these are the outer terms
- I = Inside: multiply the last term in the first bracket by the first term in the last bracket
  - Visually, these are the inner terms
- L = Last: multiply together the last terms in each bracket
It helps to put negative terms in brackets when multiplying
Simplify the final answer by collecting like terms (if there are any)

How do I expand two brackets using a grid?

You may prefer a more visual method using a grid
To expand (x + 1)(x + 3), write out the brackets as row and column headings of a grid
- They can be in either direction
- Remember to write the appropriate sign in front of each term
x
+1
x

+3
For each cell in the grid, multiply the term in the row heading by the term in the column heading
x
+1
x
x²
x
+3
3x
3
Add together all the terms inside the grid to get the answer
- x² + x + 3x + 3
Collect like terms
x² + 4x + 3

	x	+1
x	x²	x
+3	3x	3

How do I expand when there are multiple variables?

All the same rules and methods apply as when there is just one variable
Remember to only simplify like terms
For example: $(3 x + 2 y) (4 x - 6 y)$
- Expanding: $12 x^{2} - 18 x y + 8 x y - 12 y^{2}$
- The $x y$ terms can be combined
- $12 x^{2} - 10 x y - 12 y^{2}$

Worked Example

(a) Expand $(2 x - 3) (x + 4)$ .

Using FOIL, multiply together the first, outer, inner and last terms

$F O I L 2 x \times x + 2 x \times 4 + (- 3) \times x + (- 3) \times 4$

Simplify each term

$2 x^{2} + 8 x - 3 x - 12$

Collect like terms (the 8x and -3x)

$2 x^{2} + 5 x - 12$

(b) Expand $(x - 3) (3 x - 5)$ .

Using FOIL, multiply together the first, outer, inner and last terms

$F O I L x \times 3 x + x \times (- 5) + (- 3) \times 3 x + (- 3) \times (- 5)$

Simplify each term

$3 x^{2} - 5 x - 9 x + 15$

Collect like terms (the -5x and -9x)

$3 x^{2} - 14 x + 15$

Worked Example

Expand $(3 r + 2 t) (5 t - 8 r)$ .

Expand using your chosen method, here we will use a grid

	$3 r$	$+ 2 t$
$5 t$
$- 8 r$

Work out the term in each place in the grid by multiplying

	$3 r$	$+ 2 t$
$5 t$	$15 r t$	$10 t^{2}$
$- 8 r$	$- 24 r^{2}$	$- 16 r t$

So the expanded expression is

$10 t^{2} + 15 r t - 16 r t - 24 r^{2}$

The $r t$ terms can be combined

$10 t^{2} - r t - 24 r^{2}$

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Expanding Squared Brackets

How do I expand a bracket squared?

Remember that a square number is a number multiplied by itself
Write (x + 3)² as (x + 3)(x + 3) and use one of the methods above
- With FOIL: (x + 3)(x + 3) = x²+ 3x + 3x + 9
- Then collect like terms: x² + 6x + 9
Do not make the common mistake of saying (x + 3)² is x² + 3²
- This cannot be true, try substituting in x = 1
  - you would get (1 + 3)² = 4² = 16 on the left
  - but 1² + 3² = 1 + 9 = 10 on the right

Worked Example

Expand ${(2 x + 3)}^{2}$ .

Remember that the answer is not (2x)² + 3²
Rewrite the expression as two separate brackets multiplied together

$(2 x + 3) (2 x + 3)$

Using FOIL, multiply together the first, outer, inner and last terms

$F O I L 2 x \times 2 x + 2 x \times 3 + 3 \times 2 x + 3 \times 3$

Simplify each term

$4 x^{2} + 6 x + 6 x + 9$

Collect like terms (the 6x and 6x)

$4 x^{2} + 12 x + 9$

Last updated: 17 October 2024

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Previous:Expanding & Simplifying Single BracketsNext:Expanding Triple Brackets

Expanding Double Brackets (Cambridge (CIE) IGCSE Maths)

Revision Note

Expanding Two Brackets

How do I expand two brackets using FOIL?

How do I expand two brackets using a grid?

How do I expand when there are multiple variables?

Worked Example

Worked Example

Expanding Squared Brackets

How do I expand a bracket squared?

Worked Example

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Author: Mark Curtis

Author: Dan Finlay