Question

Find the square. (1/4A + 1/4B)^2

a) 1/16A^2 + 1/8AB + 1/16B^2
b) 1/4A^2 + 1/8AB + 1/4B^2

Answer 1

`(1)/(16)` A² +
`(1)/(8)` AB +
`(1)/(16)` B²

(
`(1)/(4)` A +
`(1)/(4)` B )²

= (
`(1)/(4)` A +
`(1)/(4)` B )(
`(1)/(4)` A +
`(1)/(4)` B )

expand the factors using FOIL

= (
`(1)/(4)` A )² + (2 ×
`(1)/(16)` AB + (
`(1)/(4)` B )²

=
`(1)/(16)` A² +
`(1)/(8)` AB +
`(1)/(16)` B²

Answer 2

`(a + b)^2 = a^2 + 2ab + b^2`

`((1)/(4)A + (1)/(4)B)^2 =`

`= (1)/(16)A^2 + (1)/(8)AB + (1)/(16)B^2`

Answer: A.

Another way to solve by factoring 1/4.

`((1)/(4)A + (1)/(4)B)^2 =`

`= [(1)/(4)(A + B)]^2`

`= ((1)/(4))^2(A + B)^2`

`= (1)/(16)(A^2 + 2AB + B^2)`

`= (1)/(16)A^2 + (1)/(16)2AB + (1)/(16)B)^2`

`= (1)/(16)A^2 + (1)/(8)AB + (1)/(16)B)^2`