# Lesson 10

Rewriting Quadratic Expressions in Factored Form (Part 4)

• Let’s transform more-complicated quadratic expressions into the factored form.

### Problem 1

To write $$11x^2+17x-10$$ in factored form, Diego first listed pairs of factors of -10.

$$(\underline{\hspace{.25in}}+ 5)(\underline{\hspace{.25in}} + \text-2)$$

$$(\underline{\hspace{.25in}}+ 2)(_\underline{\hspace{.25in}} + \text-5)$$

$$(\underline{\hspace{.25in}} + 10) (\underline{\hspace{.25in}} + \text-1)$$

$$(\underline{\hspace{.25in}} + 1) (\underline{\hspace{.25in}}+ \text-10)$$

1. Use what Diego started to complete the rewriting.
2. How did you know you’ve found the right pair of expressions? What did you look for when trying out different possibilities?

### Problem 2

To rewrite $$4x^2-12x-7$$ in factored form, Jada listed some pairs of factors of $$4x^2$$:

$$(2x+ \underline{\hspace{.25in}})(2x + \underline{\hspace{.25in}})$$

$$(4x + \underline{\hspace{.25in}})(1x + \underline{\hspace{.25in}})$$

Use what Jada started to rewrite $$4x^2-12x-7$$ in factored form.

### Problem 3

Rewrite each quadratic expression in factored form. Then, use the zero product property to solve the equation.

1. $$7x^2-22x+3=0$$
2. $$4x^2+x-5=0$$
3. $$9x^2-25=0$$

### Problem 4

Han is solving the equation $$5x^2+13x-6=0$$.

Here is his work:

\begin{align} 5x^2+13x-6 &= 0 \\ (5x-2)(x+3) &= 0\\x=2 \quad &\text{ or }\quad x=\text-3 \end{align}

Describe Han’s mistake. Then, find the correct solutions to the equation.

### Problem 5

A picture is 10 inches wide by 15 inches long. The area of the picture, including a frame that is $$x$$ inch thick, can be modeled by the function $$A(x) = (2x+10)(2x+15)$$.

1. Use function notation to write a statement that means: the area of the picture, including a frame that is 2 inches thick, is 266 square inches.
2. What is the total area if the picture has a frame that is 4 inches thick?
(From Unit 7, Lesson 1.)

### Problem 6

To solve the equation $$0 = 4x^2 -28x + 39$$, Elena uses technology to graph the function $$f(x) = 4x^2 -28x + 39$$. She finds that the graph crosses the $$x$$-axis at $$(1.919,0)$$ and $$(5.081,0)$$.

1. What is the name for the points where the graph of a function crosses the $$x$$-axis?
2. Use a calculator to compute $$f(1.919)$$ and $$f(5.081)$$.
3. Explain why 1.919 and 5.081 are approximate solutions to the equation $$0 = 4x^2 -28x + 39$$ and are not exact solutions.
(From Unit 7, Lesson 2.)

### Problem 7

Which equation shows a next step in solving $$9(x-1)^2=36$$ that will lead to the correct solutions?

A:

$$9(x-1) = 6 \quad \text{ or } \quad 9(x-1) = \text- 6$$

B:

$$3(x-1)=6$$

C:

$$(x-1)^2=4$$

D:

$$(9x-9)^2=36$$

(From Unit 7, Lesson 3.)

### Problem 8

Here is a description of the temperature at a certain location yesterday.

“It started out cool in the morning, but then the temperature increased until noon. It stayed the same for a while, until it suddenly dropped quickly! It got colder than it was in the morning, and after that, it was cold for the rest of the day.”

Sketch a graph of the temperature as a function of time.

Technology required. The number of people, $$p$$, who watch a weekly TV show is modeled by the equation $$p = 100,\!000 \boldcdot (1.1)^w$$, where $$w$$ is the number of weeks since the show first aired.