Expanding Two Brackets | Edexcel GCSE Maths: Foundation Revision Notes 2017

Expanding Double Brackets

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)

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

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

(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)

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

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)

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