Step by step
The question
Question: work out (2 + i)/(3 − 4i).
Remember: i is the square root of minus 1.
So i squared is minus 1.
1 Multiply by the conjugate
The bottom has an i in it.
To remove it, we use the conjugate of 3 − 4i: the same number with the sign of the i part changed.
It is 3 + 4i.
Careful!
We can't divide part by part.
The bottom 3 − 4i is one number.
First we make it a plain number.
We multiply the top and the bottom by 3 + 4i.
(3 + 4i)/(3 + 4i) is 1, so the value stays the same.
2 Multiply out
On top, each part times each part.
On the bottom too.
3 Change i² to −1
i² is minus 1.
So 4i² and −16i² change sign.
On the bottom, 12i − 12i make zero.
Cross them out.
That is why we use the conjugate.
4 Group like parts
Real parts together, i parts together.
Top: 2 + 11i.
Bottom: 25, a plain number.
No i at the bottom now.
5 Split the fraction
Each part on top is divided by 25.
So the answer is 2/25 + (11/25)i.
6 Check
Let's check: times 3 − 4i undoes divided by 3 − 4i.
Our top 2 + 11i times 3 − 4i is 50 + 25i.
That is 25 times the question's top, 2 + i, and the bottom is 25.
So the answer is right!
Let's look back.
First: multiply by the conjugate.
Then: multiply out.
Then: change i² to −1.
Then: group like parts.
Then: split the fraction.
Last: check.