# Lesson 16

### Problem 1

What number should be added to the expression $$x^2 - 15x$$ to result in an expression equivalent to a perfect square?

A:

-7.5

B:

7.5

C:

-56.25

D:

56.25

### Solution

For access, consult one of our IM Certified Partners.

### Problem 2

Noah uses the quadratic formula to solve the equation $$2x^2+3x-5=4$$. He finds $$x = \text-2.5$$ or 1. But, when he checks his answer, he finds that neither -2.5 nor 1 are solutions to the equation. Here are his steps:

$$a=2$$, $$b=3$$, $$c=\text-5$$

$$x=\frac{\text-3 \pm \sqrt{3^2 - 4 \boldcdot 2 \boldcdot \text-5}}{2 \boldcdot 2}$$

$$x=\frac{\text-3 \pm \sqrt{49}}{4}$$

$$x = \text-2.5$$ or 1

1. Explain what Noah’s mistake was.
2. Solve the equation correctly.

### Solution

For access, consult one of our IM Certified Partners.

### Problem 3

1. $$x^2-2x=\text-1$$
2. $$x^2+8x+14=23$$
3. $$x^2-15=0$$
4. $$7x^2-2x-5=0$$
5. $$2x^2+12x=8$$

### Solution

For access, consult one of our IM Certified Partners.

### Problem 4

What are the solutions to the equation $$x^2-4x=\text-3$$?

A:

$$\frac{4 \pm \sqrt{16 - 4 \boldcdot 0 \boldcdot \text-3}}{2 \boldcdot 0}$$

B:

$$\frac{4 \pm \sqrt{16 - 4 \boldcdot 1 \boldcdot \text-3}}{2 \boldcdot 1}$$

C:

$$\frac{4 \pm \sqrt{16 - 4 \boldcdot 1 \boldcdot 3}}{2 \boldcdot 1}$$

D:

$$\frac{\text-4 \pm \sqrt{16 - 4 \boldcdot 1 \boldcdot 3}}{2 \boldcdot 1}$$

### Solution

For access, consult one of our IM Certified Partners.

### Problem 5

Which expression is equivalent to $$\sqrt{\text-23}$$?

A:

$$\text-23i$$

B:

$$23i$$

C:

$$\text- i \sqrt{23}$$

D:

$$i \sqrt{23}$$

### Solution

For access, consult one of our IM Certified Partners.

(From Unit 3, Lesson 11.)

### Problem 6

Write each expression in the form $$a+bi$$, where $$a$$ and $$b$$ are real numbers.

1. $$5i^2$$
2. $$i^2 \boldcdot i^2$$
3. $$(\text-3i)^2$$
4. $$7 \boldcdot 4i$$
5. $$(5+4i) - (\text-3 + 2i)$$

### Solution

For access, consult one of our IM Certified Partners.

(From Unit 3, Lesson 12.)

### Problem 7

Let $$m=(7-2i)$$ and $$k=3i$$. Write each expression in the form $$a+bi$$, where $$a$$ and $$b$$ are real numbers.

1. $$k-m$$
2. $$k^2$$
3. $$m^2$$
4. $$k \boldcdot m$$

### Solution

For access, consult one of our IM Certified Partners.

(From Unit 3, Lesson 13.)