Simplify (3 - 2i)(2 + 5i)

Step by step

The question

Question: work out (3 − 2i)(2 + 5i).

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.

3 times 2 is 6.

3 times 5i is 15i.

−2i times 2 is −4i.

−2i times 5i is −10i².

We write all the parts.

2 Change i² to −1

i squared is minus 1.

So minus 10i² becomes 10.

Minus times minus is plus.

Now we have 6 + 15i − 4i + 10.

3 Group like parts

Real parts together, i parts together.

Real parts: 6 + 10 = 16.

i parts: 15i − 4i = 11i.

So the answer is 16 + 11i.

4 Check

Let's check: multiply the other way round, (2 + 5i) × (3 − 2i).

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

We get 16 + 11i 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.