# 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 \((3 - 5i)(\text-8 + 2i)\).

\(\text-24 + 6i - 40i + 10i^2\)

\(\text-24 + 46i - 10\)

\(\text-24 + 6i + 40i - 10i^2\)

\(\text-14 - 34i\)

\(\text-34 - 34i\)

\(\text-24 + 46i + 10\)

\(46i - 14\)

\(\text-34 + 46i\)

### 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:

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

- What could go in the blanks?
- 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.

- \(x^2=49\)
- \(x^3=49\)
- \(x^2=\text-49\)
- \(x^3=\text-49\)

### 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.

- \(2(3+2i)\)
- \(i(3+2i)\)
- \(\text-i(3+2i)\)
- \((3-2i)(3+2i)\)

### 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 |

- Describe a pattern in how each account balance changed from one year to the next.
- 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.
- Will account \(A\) ever have the same balance as account \(B\)? If so, when? Explain how you know.