# 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.

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

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

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

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

- Use what Diego started to complete the rewriting.
- 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.

- \(7x^2-22x+3=0\)
- \(4x^2+x-5=0\)
- \(9x^2-25=0\)

### 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 \(A(x) = (2x+10)(2x+15)\).

- 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.
- What is the total area if the picture has a frame that is 4 inches thick?

### 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 \((5.081,0)\).

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

### Problem 7

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

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

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

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

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

### 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.

### Problem 9

*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.

- How many people watched the show the first time it aired? Explain how you know.
- Use technology to graph the equation.
- On which week does the show first get an audience of more than 500,000 people?