Expanding Double Brackets (Edexcel IGCSE Maths)

Expanding Two Brackets

Spec point — 2.2E

How do I expand two 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

    • x(x + 1) + 3(x + 1)
    • x×x + 1x + 3x + 3
    • x² + 4x + 3

Using a grid

• You can use a grid for a visual way to expand brackets

• To expand (x + 1)(x + 3), write out the brackets as row and column headings of a grid | | x | +1 | |---|---|---| | x | | | | +3 | | |

  • They can be in either direction
  • Remember to write the appropriate sign in front of each term

• 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

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 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: $12x^{2}-18xy+8xy-12y^{2}$
  • The $xy$ terms can be combined
  • $12x^{2}-10xy-12y^{2}$

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

Answer:

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

F O I L > circle 2 x x x + circle 2 x x 4 + circle ( - 3 ) x x + circle ( - 3 ) x 4

Simplify each term

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

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

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

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

Answer:

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

F O I L > circle x x 3 x + circle x x ( - 5 ) + circle ( - 3 ) x 3 x + circle ( - 3 ) x ( - 5 )

Simplify each term

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

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

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

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

Answer:

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

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

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

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

So the expanded expression is

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

The $rt$ terms can be combined

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

Expanding Squared Brackets

Spec point — 2.2E

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

Exam Hint Do not make the common mistake of saying $( x + y )^{2}$ is the same as $x^{2}+y^{2}$. In general, this is not true.

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

Answer:

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 > circle 2 x x 2 x + circle 2 x x 3 + circle 3 x 2 x + circle 3 x 3

Simplify each term

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

Collect like terms (the 6x and 6x)

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