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Grade 11 · Math

Multiply a pair of complex conjugates

Use i² = −1 and conjugate structure to find a real product.

6 authored questions + worked example

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Worked example

Let i2 = −1. Find (2 + 3i)(2 − 3i).

Answer & why

Answer: 13

The middle terms cancel. 22 − (3i)2 = 4 − 9(−1) = 13.

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All questions

All 6 authored questions for this skill.

Question 1

Let i2 = −1. Find (4 + 1i)(4 − 1i).

  1. 17
  2. 19
  3. 16
  4. 18
A hint

Use the difference-of-squares identity, then replace i2 with −1.

Answer & why

Answer: 17

The middle terms cancel. 42 − (1i)2 = 16 − 1(−1) = 17.

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Question 2

Let i2 = −1. Find (5 + 2i)(5 − 2i).

  1. 31
  2. 28
  3. 29
  4. 30
A hint

Use the difference-of-squares identity, then replace i2 with −1.

Answer & why

Answer: 29

The middle terms cancel. 52 − (2i)2 = 25 − 4(−1) = 29.

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Question 3

Let i2 = −1. Find (2 + 5i)(2 − 5i).

  1. 29
  2. 28
  3. 30
  4. 31
A hint

Use the difference-of-squares identity, then replace i2 with −1.

Answer & why

Answer: 29

The middle terms cancel. 22 − (5i)2 = 4 − 25(−1) = 29.

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Question 4

Let i2 = −1. Find (6 + 3i)(6 − 3i).

  1. 47
  2. 46
  3. 45
  4. 44
A hint

Use the difference-of-squares identity, then replace i2 with −1.

Answer & why

Answer: 45

The middle terms cancel. 62 − (3i)2 = 36 − 9(−1) = 45.

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Question 5

Let i2 = −1. Find (2 + 3i)(2 − 3i).

  1. 13
  2. 12
  3. 15
  4. 14
A hint

Use the difference-of-squares identity, then replace i2 with −1.

Answer & why

Answer: 13

The middle terms cancel. 22 − (3i)2 = 4 − 9(−1) = 13.

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Question 6

Let i2 = −1. Find (3 + 4i)(3 − 4i).

  1. 25
  2. 24
  3. 26
  4. 27
A hint

Use the difference-of-squares identity, then replace i2 with −1.

Answer & why

Answer: 25

The middle terms cancel. 32 − (4i)2 = 9 − 16(−1) = 25.

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