For i = √-1, which of the following equals (3 + 2i)(1

Step by step

The question

Question: work out (3 + 2i)(1 − i).

(3 + 2i)(1 − i)

The numbers we know are called real numbers.

No real number times itself is negative.

3 times 3 is 9, and minus 3 times minus 3 is 9 too.

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

A square root times itself gives back what is under it.

So i squared is minus 1.

3 × 3 = 9

(−3) × (−3) = 9

i = √(−1)

i² = √(−1) × √(−1)

= −1

There are four choices.

First we solve it, the normal way.

Then we pick.

1 Multiply each part

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

3 times 1 is 3.

3 times −i is −3i.

2i times 1 is 2i.

2i times −i is −2i².

3 × 1, 3 × (−i), 2i × 1, 2i × (−i)

We write all the parts.

= 3 − 3i + 2i − 2i²

2 Change i² to −1

i squared is minus 1.

So minus 2i² becomes 2.

Minus times minus is plus.

−2i² = −2 × (−1)

= 2

Now we have 3 − 3i + 2i + 2.

= 3 − 3i + 2i + 2

3 Group like parts

Real parts together, i parts together.

Real parts: 3 + 2 = 5.

i parts: −3i + 2i = −i.

3 + 2 = 5

−3i + 2i = −i

So the answer is 5 − i.

= 5 − i

4 Check

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

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

We get 5 − i again.

So the answer is right!

Check: (1 − i) × (3 + 2i) = 3 + 2i − 3i − 2i²

= 3 + 2i − 3i + 2 = 5 − i

5 Pick the choice

Our answer is 5 − i.

That is choice B.

B) 5 − i

6 Why not the others

A is 1 − i.

Wrong: i² is −1, so −2i² is +2, not −2.

A) 1 − i ✗ i² is −1, so −2i² is +2, not −2

C is 5 − 5i.

Wrong: 2i × 1 is +2i, not −2i.

C) 5 − 5i ✗ 2i × 1 is +2i, not −2i

D is 5 + 5i.

Wrong: 3 × (−i) is −3i, not +3i.

D) 5 + 5i ✗ 3 × (−i) is −3i, not +3i

Let's look back.

First: multiply each part.

Then: change i² to −1.

Then: group like parts.

Then: check.

Then: pick the choice.

Last: why not the others.

Answer