# Lesson 18

The Quadratic Formula and Complex Solutions

### Problem 1

Clare solves the quadratic equation $$4x^2+12x+58=0$$, but when she checks her answer, she realizes she made a mistake. Explain what Clare's mistake was.

\begin{align} x &= \frac{\text-12 \pm \sqrt{12^2 - 4 \boldcdot 4 \boldcdot 58}}{2 \boldcdot 4} \\ x &= \frac{\text-12 \pm \sqrt{144-928}}{8} \\ x &= \frac{\text-12 \pm \sqrt{\text-784}}{8} \\ x &= \frac{\text-12 \pm 28i}{8} \\ x &= \text-1.5 \pm 28i \\ \end{align}

### Problem 2

Write in the form $$a+bi$$, where $$a$$ and $$b$$ are real numbers:

1. $$\frac{5 \pm \sqrt{\text-4}}{3}$$
2. $$\frac{10 \pm \sqrt{\text-16}}{2}$$
3. $$\frac{\text-3 \pm \sqrt{\text-144}}{6}$$

### Problem 3

Priya is using the quadratic formula to solve two different quadratic equations.

For the first equation, she writes $$x= \frac{4 \pm \sqrt{16-72}}{12}$$

For the second equation, she writes $$x=\frac{8 \pm \sqrt{64-24}}{6}$$

Which equation(s) will have real solutions? Which equation(s) will have non-real solutions? Explain how you know.

### Problem 4

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

1. $$x^2=25$$
2. $$x^3 = 27$$
3. $$x^2=12$$
4. $$x^3=12$$

### Solution

(From Unit 3, Lesson 8.)

### Problem 5

Kiran is solving the equation $$\sqrt{x+2} - 5 = 11$$ and decides to start by squaring both sides. Which equation results if Kiran squares both sides as his first step?

A:

$$x + 2 - 25 = 121$$

B:

$$x + 2 + 25 = 121$$

C:

$$x+2 - 10\sqrt{x+2} + 25 = 121$$

D:

$$x+2 + 10\sqrt{x+2} + 25 = 121$$

### Solution

(From Unit 3, Lesson 9.)

### Problem 6

Plot each number on the real or imaginary number line.

1. $$\text-\sqrt{4}$$
2. $$\sqrt{\text-1}$$
3. $$3\sqrt{4}$$
4. $$\text-3\sqrt{\text-1}$$
5. $$4\sqrt{\text-1}$$
6. $$2\sqrt{2}$$