# Lesson 17

Completing the Square and Complex Solutions

• Let’s find complex solutions to quadratic equations by completing the square.

### Problem 1

Find the solution or solutions to each equation.

1. $$x^2+0.5x-14=0$$
2. $$x^2+12x+36=0$$
3. $$x^2-3x+8=0$$
4. $$x^2+4=0$$

### Problem 2

Which describes the solutions to the equation $$x^2+7=0$$?

A:

One real solution

B:

Two real solutions

C:

One non-real solution

D:

Two non-real solutions

### Problem 3

Explain how you know $$\sqrt{3x+2}=\text-16$$ has no solutions.

(From Unit 3, Lesson 7.)

### Problem 4

Determine the number of real solutions and non-real solutions to each equation. Use the graphs; don't do any calculations to find the solutions.

1. $$x^2-6x+7=0$$
2. $$3x^2+2x+1=0$$
3. $$\text-x^2-3x+2=0$$
4. $$x^2-6x+7=\text-2$$
5. $$\text-x^2-3x+2=6$$
6. $$3x^2+2x+1=2$$

### Problem 5

1. Write $$(5-5i)^2$$ in the form $$a+bi$$, where $$a$$ and $$b$$ are real numbers.
2. Write $$(5-5i)^4$$ in the form $$a+bi$$, where $$a$$ and $$b$$ are real numbers.
(From Unit 3, Lesson 14.)

### Problem 6

What number $$n$$ makes this equation true?

$$x^2+11x+\frac{121}{4} = (x+n)^2$$

A:

$$\frac{11}{4}$$

B:

$$\frac{11}{2}$$

C:

11

D:

$$\frac{121}{4}$$

(From Unit 3, Lesson 16.)