# Lesson 15

Working Backwards

Let's use what we've learned about multiplying complex numbers.

### Problem 1

Select all the expressions that are equivalent to $$(3 - 5i)(\text-8 + 2i)$$.

A:

$$\text-24 + 6i - 40i + 10i^2$$

B:

$$\text-24 + 46i - 10$$

C:

$$\text-24 + 6i + 40i - 10i^2$$

D:

$$\text-14 - 34i$$

E:

$$\text-34 - 34i$$

F:

$$\text-24 + 46i + 10$$

G:

$$46i - 14$$

H:

$$\text-34 + 46i$$

### Problem 2

Explain or show how to write $$(20 - i)(8 + 4i)$$ in the form $$a+bi$$, where $$a$$ and $$b$$ are real numbers.

### Problem 3

Without going through all the trouble of writing the left side in the form $$a+bi$$, how could you tell that this equation is false?

$$(\text-9 + 2i)(10 - 13i) =\text-68 - 97i$$

### Problem 4

Andre spilled something on his math notebook and some parts of the problems he was working on were erased. Here is one of the problems:

$$(\hspace{1cm} - 2i)(\hspace{1cm} + 2i) = \hspace{1cm} - 10i$$

1. What could go in the blanks?
2. Could other numbers work, or is this the only possibility? Explain your reasoning.

### Problem 5

Find the exact solution(s) to each of these equations, or explain why there is no solution.

1. $$x^2=49$$
2. $$x^3=49$$
3. $$x^2=\text-49$$
4. $$x^3=\text-49$$
(From Unit 3, Lesson 8.)

### Problem 6

Write each expression in the form $$a+bi$$, where $$a$$ and $$b$$ are real numbers. Optionally, plot $$3+2i$$ in the complex plane. Then plot and label each of your answers.

1. $$2(3+2i)$$
2. $$i(3+2i)$$
3. $$\text-i(3+2i)$$
4. $$(3-2i)(3+2i)$$
(From Unit 3, Lesson 13.)

### Problem 7

The table shows two investment account balances growing over time.

time
(years since 2000)
account $$A$$
(thousands of dollars)
account $$B$$
(thousands of dollars)
0 5 10
1 5.1 10.15
2 5.2 10.3
3 5.3 10.45
4 5.4 10.6
1. Describe a pattern in how each account balance changed from one year to the next.
2. Define the amount of money, in thousands of dollars, in accounts $$A$$ and $$B$$ as functions of time $$t$$, where $$t$$ is years since 2000, using function notation.
3. Will account $$A$$ ever have the same balance as account $$B$$? If so, when? Explain how you know.
(From Unit 1, Lesson 10.)