# Lesson 8

Rewriting Quadratic Expressions in Factored Form (Part 3)

### Problem 1

Match each quadratic expression given in factored form with an equivalent expression in standard form. One expression in standard form has no match.

### Problem 2

Both $$(x-3)(x+3)$$ and $$(3-x)(3+x)$$ contain a sum and a difference and have only 3 and $$x$$ in each factor.

If each expression is rewritten in standard form, will the two expressions be the same? Explain or show your reasoning.

### Problem 3

1. Show that the expressions $$(5+1)(5-1)$$ and $$5^2-1^2$$ are equivalent.
2. The expressions $$(30-2)(30+2)$$ and $$30^2-2^2$$ are equivalent and can help us find the product of two numbers. Which two numbers are they?
3. Write $$94\boldcdot106$$ as a product of a sum and a difference, and then as a difference of two squares. What is the value of $$94\boldcdot106$$?

### Problem 4

Write each expression in factored form. If not possible, write “not possible.”

1. $$x^2 - 144$$
2. $$x^2 + 16$$
3. $$25 - x^2$$
4. $$b^2 - a^2$$
5. $$100 + y^2$$

### Problem 5

What are the solutions to the equation $$(x-a)(x+b)=0$$?

A:

$$a$$ and $$b$$

B:

$$\text-a$$ and $$\text-b$$

C:

$$a$$ and $$\text-b$$

D:

$$\text-a$$ and $$b$$

### Solution

(From Unit 7, Lesson 4.)

### Problem 6

Create a diagram to show that $$(x-3)(x-7)$$ is equivalent to $$x^2-10x+21$$.

### Solution

(From Unit 7, Lesson 6.)

### Problem 7

Select all the expressions that are equivalent to $$8 - x$$.

A:

$$x - 8$$

B:

$$8 + (\text-x)$$

C:

$$\text-x - (\text -8)$$

D:

$$\text-8 + x$$

E:

$$x - (\text-8)$$

F:

$$x +(\text -8)$$

G:

$$\text-x + 8$$

### Solution

(From Unit 7, Lesson 6.)

### Problem 8

Mai fills a tall cup with hot cocoa, 12 centimeters in height. She waits 5 minutes for it to cool. Then, she starts drinking in sips, at an average rate of 2 centimeters of height every 2 minutes, until the cup is empty.

The function $$C$$ gives the height of hot cocoa in Mai’s cup, in centimeters, as a function of time, in minutes.

1. Sketch a possible graph of $$C$$. Be sure to include a label and a scale for each axis.
2. What quantities do the domain and range represent in this situation?
3. Describe the domain and range of $$C$$.

### Solution

(From Unit 4, Lesson 11.)

### Problem 9

Two bacteria populations are measured at the same time. One bacteria population, $$p$$, is modeled by the equation $$p = 250,\!000 \boldcdot \left(\frac{1}{2} \right)^d$$, where $$d$$ is the number of days since it was first measured. The second bacteria population, $$q$$, is modeled by the equation $$q = 500,\!000 \boldcdot \left(\frac{1}{3}\right)^d$$.

Which statement is true about the two populations?

A:

The second population will always be larger than the first.

B:

Both populations are increasing.

C:

The second bacteria population decreases more rapidly than the first.

D:

When initially measured, the first population is larger than the second.