Rewriting Quadratic Expressions (Part 4): IM Alg1.7.10
[url=https://im.kendallhunt.com/HS/teachers/1/7/10/preparation.html]“Rewriting Quadratic Expressions in Factored Form (Part 4)”[/url] from IM Algebra I © 2019 [url=https://www.illustrativemathematics.org/]Illustrative Mathematics[/url]. Licensed under the [url=https://creativecommons.org/licenses/by/4.0/]Creative Commons Attribution 4.0[/url] license.
Table of Contents
- Alg1.7.10 Lesson
- IM Alg1.7.10 Lesson: Rewriting Quadratic Expressions in Factored Form (Part 4)
- Alg1.7.10 Practice
- IM Alg1.7.10 Practice: Rewriting Quadratic Expressions in Factored Form (Part 4)
IM Alg1.7.10 Lesson: Rewriting Quadratic Expressions in Factored Form (Part 4)
Which one doesn’t belong?
[size=150]A. [math](x+4)(x-3)[/math][br][br]B.[math]3x^2-8x+5[/math][br][br]C. [math]x^2-25[/math][br][br]D. [math]x^2+2x+3[/math][br][br][/size] Explain your reasoning.
Each row in each table has a pair of equivalent expressions. Complete the tables. If you get stuck, try drawing a diagram in the applet below the table.
[size=150]Here are three quadratic equations, each with two solutions. Find both solutions to each equation, using the zero product property somewhere along the way. Show each step in your reasoning.[/size][br][br][math]x^2=6x[/math]
[math]x\left(x+4\right)=x+4[/math]
[math]2x\left(x-1\right)+3x-3=0[/math]
[size=150]An engineer is designing a fountain that shoots out drops of water. The nozzle from which the water is launched is 3 meters above the ground. It shoots out a drop of water at a vertical velocity of 9 meters per second.[br][br]Function [math]h[/math] models the height in meters, [math]h[/math], of a drop of water [math]t[/math] seconds after it is shot out from the nozzle. The function is defined by the equation [math]h\left(t\right)=-5t^2+9t+3[/math].[/size][br][br][size=150]How many seconds until the drop of water hits the ground?[/size][br]Write an equation that we could solve to answer the question.
Try to solve the equation by writing the expression in factored form and using the zero product property.[br]
Try to solve the equation by graphing the function using graphing technology.
Explain how you found the solution.[br]
Here is a clever way to think about quadratic expressions that would make it easier to rewrite them in factored form.
[math]9x^2+21x+10 \\ [br](3x)^2+7(3x)+10 \\ [br]N^2+7N+10\\ [br](N+2)(N+5) \\[br]3x+2)(3x+5)[/math][br][br]Use the distributive property to expand [math]\left(3x+2\right)\left(3x+5\right)[/math]. Show your reasoning and write the resulting expression in standard form. Is it equivalent to [math]9x^2+21x+10[/math]?
Study the method and make sense of what was done in each step. Make a note of your thinking and be prepared to explain it.[br]
Try the method to write each of these expressions in factored form:[br][br][list][*][math]4x^2+28x+45[/math][/*][/list]
[list][*][math]25x^2-35x+6[/math][/*][/list]
[size=100]You have probably noticed that the coefficient of the squared term in all of the previous examples is a perfect square. [/size]What if that coefficient is not a perfect square?
Here is an example of an expression whose squared term has a coefficient that is not a squared term.[br][br][math]5x^2+17x+6 \\ [br]\frac{1}{5} \cdot 5 \cdot (5x^2 + 17x + 6)\\ [br]\frac{1}{5} (25x^2 + 85x + 30) \\[br]\frac{1}{5} ((5x)^2 + 17 (5x) + 30)\\[br]\frac{1}{5} (N^2 + 17N + 30)\\[br]\frac{1}{5} (N+15)(N+2) \\ [br]\frac{1}{5} (5x+15)(5x+2) \\[br](x+3)(5x+2)[/math][br][br]Use the distributive property to expand [math]\left(x+3\right)\left(5x+2\right)[/math]. Show your reasoning and write the resulting expression in standard form. Is it equivalent to [math]5x^2+17x+6[/math]?
Study the method and make sense of what was done in each step and why. Make a note of your thinking and be prepared to explain it.[br]
Try the method to write each of these expressions in factored form:[br][list][*][math]3x^2+16x+5[/math][/*][/list]
[list][*][math]10x^2-41x+4[/math][/*][/list]
IM Alg1.7.10 Practice: Rewriting Quadratic Expressions in Factored Form (Part 4)
[size=150]To write [math]11x^2+17x-10[/math] in factored form, Diego first listed pairs of factors of -10.[br][/size][br][math](\underline{\hspace{.25in}}+ 5)(\underline{\hspace{.25in}} + \text{-}2)[/math][br][math](\underline{\hspace{.25in}}+ 2)(\underline{\hspace{.25in}} + \text{-}5)[/math][br][math](\underline{\hspace{.25in}} + 10) (\underline{\hspace{.25in}} + \text{-}1)[/math][br][math](\underline{\hspace{.25in}} + 1) (\underline{\hspace{.25in}}+ \text{-}10)[/math][br][br][size=100]Use what Diego started to complete the rewriting.[/size]
How did you know you’ve found the right pair of expressions? What did you look for when trying out different possibilities?[br]
[size=150]To rewrite [math]4x^2-12x-7[/math] in factored form, Jada listed some pairs of factors of [math]4x^2[/math]:[br][/size][br][math](2x+ \underline{\hspace{.25in}})(2x + \underline{\hspace{.25in}})[/math][br][math](4x + \underline{\hspace{.25in}})(1x + \underline{\hspace{.25in}})[/math][br][size=100][br]Use what Jada started to rewrite [math]4x^2-12x-7[/math] in factored form.[/size]
Rewrite each quadratic expression in factored form. Then, use the zero product property to solve the equation.
[math]7x^2-22x+3=0[/math]
[math]4x^2+x-5=0[/math]
[math]9x^2-25=0[/math]
[size=150]Han is solving the equation [math]5x^2+13x-6=0[/math]. Here is his work:[/size][br][br][math]\begin{align} 5x^2+13x-6 &= 0 \\ (5x-2)(x+3) &= 0\\x=2 \quad &\text{ or }\quad x=\text-3 \end{align}[/math][br][br][size=100]Describe Han’s mistake. Then, find the correct solutions to the equation.[/size]
[size=150]A picture is 10 inches wide by 15 inches long. The area of the picture, including a frame that is [math]x[/math] inch thick, can be modeled by the function [math]A\left(x\right)=\left(2x+10\right)\left(2x+15\right)[/math].[br][/size][br][size=100]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.[/size]
What is the total area if the picture has a frame that is 4 inches thick?
[size=150]To solve the equation [math]0=4x^2-28x+39[/math], Elena uses technology to graph the function [math]f\left(x\right)=4x^2-28x+39[/math]. She finds that the graph crosses the [math]x[/math]-axis at [math]\left(1.919,0\right)[/math] and [math]\left(5.081,0\right)[/math].[/size][br][br][size=100]What is the name for the points where the graph of a function crosses the [/size][math]x[/math]-axis?
Use a calculator to compute [math]f\left(1.919\right)[/math]. You may use the applet below.[br]
Use a calculator to compute [math]f\left(5.081\right)[/math]. You may use the applet below.[br]
Explain why 1.919 and 5.081 are approximate solutions to the equation [math]0=4x^2-28x+39[/math] and are not exact solutions.[br]
[size=150]Which equation shows a next step in solving [math]9\left(x-1\right)^2=36[/math] that will lead to the correct solutions?[/size]
Here is a description of the temperature at a certain location yesterday.
[size=150]“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.”[/size][br]
Sketch a graph of the temperature as a function of time.
[size=150]The number of people, [math]p[/math], who watch a weekly TV show is modeled by the equation [math]p=100,000\cdot\left(1.1\right)^w[/math], where [math]w[/math] is the number of weeks since the show first aired.[/size][br][br]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?[br]