Question

How do you simplify `(9-i)/(2-i)`?

Answer 1

`(19 + 7i)/5` or `19/5 + (7i)/5`

Explanation:

`(9-i)/(2-i)`

Multiply both numerator and denominator by the denominator's conjugate (2+i) to remove the `i` from the denominator:
`(9-i)/(2-i) color(blue)(*(2+i)/(2+i))`

Simplify using FOIL:
`(18 * +9i + -2i + -i^2)/(4 + 2i + -2i + -i^2)`

Combine like terms. We also know that `i^2` is `-1`, so:
`(18 + 7i - (-1))/(4 - (-1))`

Simplify:
`(18 + 7i + 1)/(4 + 1)`

`(19 + 7i)/5`

Hope this helps!