Question

How do you simplify `(3+2i)/(2+i)`?

Answer 1

Multiply both the top and bottom by the conjugate of the denominator, `(2-i)`, and simplify to get `8/5-1/5i`

Explanation:

Starting with `(3+2i)/(2+i)`, we can get `i` out of the denominator by multiplying both the numerator and denominator by the "conjugate" of the denominator, which is just the denominator with the sign switched in the middle:
`(3+2i)/(2+i)*(2-i)/(2-i)`
Multiply and simplify (remember that `i=sqrt(-1)` so `i^2=-1`) to get

`(6-3i+4i-2i^2)/(4-i^2)=(6+i-2(-1))/(4-(-1))=(6+i+2)/5=(8+i)/5`

Finally, your teacher may or may not care about this, but "standard form" for a complex number is `a+-bi`, where `a` is the Real number part. Put your answer in this format by breaking up the numerator:
`(8+i)/5=8/5-1/5i`