Expanding Brackets and Simplifying

a(b + c) = ab + ac

a(b + c + d + ...) = ab + ac + ad + ...

Remember that ab = a \times b.

Lets look at some examples:

Example

Expand and simplify the following:

i) x(x + 2)

ii) 4x(2x + 3y)

iii) 5x(4x - a + b)

iv) 3(x + 2 + y)

v) 7(x + y + xy)

i) x(x + 2) = { x }^{ 2 } + 2x

ii) 4x(2x + 3y) = { 8x }^{ 2 } + 12xy

iii) 5x(4x - a + b) = { 20x }^{ 2 } - 5ax + 5bx

iv) 3(x + 2 + y) = 3x + 6 + 3y

v) 7(x + y +xy) = 7x + 7y +7xy


Example

Expand and simplify the following:

i) 3(x + y) + 2(x - y)

ii) 4(x + 2y) - (x + y)

iii) 3(x + 4) - 3(x-5)

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

v) x + x(2x + 3) - 7x + 2

i) 3(x + y) + 2(x - y) = 3x + 3y + 2x - 2y

= 5x + y

ii) 4(x + 2y) - (x + y) = 4x + 8y - x - y

= 3x + 7y

iii) 3(x + 4) - 3(x - 5) = 3x + 12 - 3x + 15

= 27

iv) (x + 2) - (x + 3) + (3x + 5) = x + 2 - x - 3 + 3x + 5

= 3x + 4

v) x + x(2x + 3) - 7x + 2= x + { 2x }^{ 2 } + 3x - 7x + 2

= { 2x }^{ 2 } - 3x + 2