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

Step by step

The question

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

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 1 is 1.

1 times 2i is 2i.

2i times 1 is 2i.

2i times 2i is 4i².

We write all the parts.

2 Change i² to −1

i squared is minus 1.

So 4i² becomes −4.

Now we have 1 + 2i + 2i − 4.

3 Group like parts

Real parts together, i parts together.

Real parts: 1 − 4 = −3.

i parts: 2i + 2i = 4i.

So the answer is −3 + 4i.

4 Check

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

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

We get −3 + 4i 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.