Step by step
The question
Question: work out (7 − i)/(2 + i).
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 2 + i: the same number with the sign of the i part changed.
It is 2 − i.
Careful!
We can't divide part by part.
The bottom 2 + i is one number.
First we make it a plain number.
We multiply the top and the bottom by 2 − i.
(2 − i)/(2 − i) 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 i² and −i² change sign.
On the bottom, −2i + 2i make zero.
Cross them out.
That is why we use the conjugate.
4 Group like parts
Real parts together, i parts together.
Top: 13 − 9i.
Bottom: 5, a plain number.
No i at the bottom now.
5 Split the fraction
Each part on top is divided by 5.
So the answer is 13/5 − (9/5)i.
6 Check
Let's check: times 2 + i undoes divided by 2 + i.
Our top 13 − 9i times 2 + i is 35 − 5i.
That is 5 times the question's top, 7 − i, and the bottom is 5.
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.