Simplify (1 + 2i)(3 + i)

Step by step

The question

Question: work out (1 + 2i)(3 + i).

No real number times itself is negative.

So we use a new number: i, the square root of minus 1.

So i squared is minus 1.

1 Multiply each part

Each part of the first bracket times each part of the second.

1 times 3 is 3.

1 times i is i.

2i times 3 is 6i.

2i times i is 2i².

We write all the parts.

2 Change i² to −1

i squared is minus 1.

So 2i² becomes −2.

Now we have 3 + i + 6i − 2.

3 Group like parts

Real parts together, i parts together.

Real parts: 3 − 2 = 1.

i parts: i + 6i = 7i.

So the answer is 1 + 7i.

4 Check

Let's check: multiply the other way round, (3 + i) × (1 + 2i).

Each part times each part again, and i² is −1.

We get 1 + 7i again.

So the answer is right!

Let's look back.

First: multiply each part.

Then: change i² to −1.

Then: group like parts.

Last: check.