Simplify (2 + i)(2 - i)

Step by step

The question

Question: work out (2 + i)(2 − 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.

2 times 2 is 4.

2 times −i is −2i.

i times 2 is 2i.

i times −i is −i².

We write all the parts.

2 Change i² to −1

i squared is minus 1.

So minus i² becomes 1.

Minus times minus is plus.

Now we have 4 − 2i + 2i + 1.

3 Group like parts

Real parts together, i parts together.

Real parts: 4 + 1 = 5.

i parts: −2i + 2i = 0.

So the answer is 5.

The i parts cancel!

4 Check

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

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

We get 5 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.