Question

How can I factor these complex conjuages? a^2 + b^2 and a^2 - b

Answer 1

`\boxed{(a+ib)(a-ib)}`

`\boxed{a^2+i^2b}`

Explanation:

`a^2 + b^2`

Rewrite expression.

`a^2- (-1)b^2`

Use identity :
`-1=i^2`

`a^2- i^2 b^2`

Factor out square.

`a^2-(ib)^2`

Apply difference of two squares formula :
`a^2-b^2 =(a+b)(a-b)`

`(a+ib)(a-ib)`

`a^2-b`

Rewrite expression.

`a^2+(-1)b`

Use identity :
`-1=i^2`

`a^2+i^2b`

Answer 2

1)
`(a+ib)(a-ib)`

2)
`a^2+i^2b`

Explanation:

1)
`a^2+b^2`

=>
`a^2 - (-1)b^2` (We know that -1 =
`i^2` )

=>
`a^2-i^2b^2`

=>
`(a)^2-(ib)^2`

Using Formula
`a^2 -b^2 = (a+b)(a-b)`

=>
`(a+ib)(a-ib)`

2)
`a^2-b`

=>
`a^2+(-1)b` (We know that -1 =
`i^2` )

=>
`a^2+i^2b` (It cannot be simplified further)