Expanding Double Brackets

Lets look at some examples where we expand double brackets, which means to write two brackets next to each other and multiply them together. In this case, every term in the first bracket have to be multiplied by every term in the second bracket.

Example

Expand and simplify:

\left(x+3\right) \left(x+5\right)

\left(x+3\right) \left(x+5\right) = x\left(x+5\right)+ 3\left(x+5\right)

= x^{2} + 5x + 3x + 15

= x^{2} + 8x + 15


Example

Expand and multiply the following:

A) (x + 3) (x + 8)

B) (x + 3) (x - 8)

C) (2x - 1) (3x + 1)

D) (5x + 3) (2x - 5)

E) (3x - 1) (3x + 1)

F) (a + b) (a - b)

G)(a + b) (a + b)

H) (a - b) (a - b)

A) (x + 3) (x + 8) = { x }^{ 2 } + 8x + 3x + 24

= { x }^{ 2 } + 11x + 24

 

B) (x + 3) (x - 8) = { x }^{ 2 } - 8x + 3x - 24

= { x }^{ 2 } - 5x - 24

 

C) (2x - 1) (3x + 1) = { 6x }^{ 2 } + 2x - 3x - 1

= { 6x }^{ 2 } - x - 1

 

D) (5x + 3) (2x - 5) = { 10x }^{ 2 } - 25x + 6x - 15

= { 10x }^{ 2 } - 19x - 15

 

E) (3x - 1) (3x + 1) = { 9x }^{ 2 } +3x - 3x - 1

= { 9x }^{ 2 } - 1

 

F) (a + b) (a - b) = { a }^{ 2 } - ab + ab - { b }^{ 2 }

= { a }^{ 2 } - { b }^{ 2 }

 

G) (a + b) (a + b) = { a }^{ 2 } + ab + ab + { b }^{ 2 }

= { a }^{ 2 } + 2ab + { b }^{ 2 }

 

H) (a - b) (a - b) = { a }^{ 2 } - ab - ab + { b }^{ 2 }

= { a }^{ 2 } - 2ab + { b }^{ 2 }


The last three expansions are worth memorising. They will be helpful.

(a + b) (a + b) = { a }^{ 2 } + 2ab + { b }^{ 2 }

(a + b) (a - b) = { a }^{ 2 } - { b }^{ 2 }

(a - b) (a - b) = { a }^{ 2 } - 2ab + { b }^{ 2 }