Rewriting Quadratic Expressions (Part 3): IM Alg1.7.8
[url=https://im.kendallhunt.com/HS/teachers/1/7/8/preparation.html]“Rewriting Quadratic Expressions in Factored Form (Part 3)”[/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
- Alg1.7.8 Lesson
- IM Alg1.7.8 Lesson: Rewriting Quadratic Expressions in Factored Form (Part 3)
- Alg1.7.8 Practice
- IM Alg1.7.8 Practice: Rewriting Quadratic Expressions in Factored Form (Part 3)
IM Alg1.7.8 Lesson: Rewriting Quadratic Expressions in Factored Form (Part 3)
Find each product mentally.
[math]9\cdot11[/math]
[math]19\cdot21[/math]
[math]99\cdot101[/math]
[math]109\cdot101[/math]
Clare claims that [math](10+3)(10-3)[/math] is equivalent to [math]10^2-3^2[/math] and [math](20+1)(20-1)[/math] is equivalent to [math]20^2-1^2[/math]. Do you agree? Show your reasoning.
Use your observations from the first question and evaluate [math]\left(100+5\right)\left(100-5\right)[/math]. Show your reasoning.[br]
Check your answer by computing [math]105\cdot95[/math].[br]
Is [math]\left(x+4\right)\left(x-4\right)[/math] equivalent to [math]x^2-4^2[/math]? Support your answer without a diagram.
Support your answer with a diagram.
Is (x+4)² equivalent to x²+4²? Support your answer, either with or without a diagram.
Explain how your work in the previous questions can help you mentally evaluate [math]22\cdot18[/math] and [math]45\cdot35[/math].[br]
[size=150][size=100]Here is a shortcut that can be used to mentally square any two-digit number. Let’s take [math]83^2[/math], for example.[br][list][*]83 is [math]80+3[/math].[/*][*]Compute [math]80^2[/math] and [math]3^2[/math], which give 6,400 and 9. Add these values to get 6,409.[/*][*]Compute [math]80\cdot3[/math], which is 240. Double it to get 480.[/*][*]Add 6,409 and 480 to get 6,889.[/*][/list][/size][br][/size]Try using this method to find the squares of some other two-digit numbers. (With some practice, it is possible to get really fast at this!) Then, explain why this method works.[br]
Each row has a pair of equivalent expressions. Complete the table. If you get stuck, consider drawing a diagram. (Heads up: one of them is impossible.)
IM Alg1.7.8 Practice: Rewriting Quadratic Expressions in Factored Form (Part 3)
Match each quadratic expression given in factored form with an equivalent expression in standard form. One expression in standard form has no match.
[size=150]Both [math]\left(x-3\right)\left(x+3\right)[/math] and [math]\left(3-x\right)\left(3+x\right)[/math] contain a sum and a difference and have only 3 and [math]x[/math] in each factor.[/size][br][br]If each expression is rewritten in standard form, will the two expressions be the same? Explain or show your reasoning.
[size=150]Show that the expressions [math]\left(5+1\right)\left(5-1\right)[/math] and [math]5^2-1^2[/math] are equivalent.[/size][br]
[size=150]The expressions [math]\left(30-2\right)\left(30+2\right)[/math] and [math]30^2-2^2[/math] are equivalent and can help us find the product of two numbers. [br][/size][br]Which two numbers are they?[br]
[size=150]Write [math]94\cdot106[/math] as a product of a sum and a difference, and then as a difference of two squares. [/size][br][br]What is the value of [math]94\cdot106[/math]?
Write each expression in factored form. If not possible, write “not possible.”
[math]x^2-144[/math]
[math]x^2+16[/math]
[math]25-x^2[/math]
[math]b^2-a^2[/math]
[math]100+y^2[/math]
[size=150]What are the solutions to the equation [math]\left(x-a\right)\left(x+b\right)=0[/math]?[/size]
Create a diagram to show that (x-3)(x-7) is equivalent to x²-10x+21.
[size=150]Select [b]all[/b] the expressions that are equivalent to [math]8-x[/math].[/size]
[size=150]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.[br][br]The function [math]C[/math] gives the height of hot cocoa in Mai’s cup, in centimeters, as a function of time, in minutes.[/size][br]Sketch a possible graph of [math]C[/math] in the applet below. Be sure to include a label and a scale for each axis.
What quantities do the domain and range represent in this situation?
Describe the domain and range of [math]C[/math].
[size=150]One bacteria population, [math]p[/math], is modeled by the equation [math]p=250,000\cdot\left(\frac{1}{2}\right)^d[/math], where [math]d[/math] is the number of days since it was first measured.[br][br]A second bacteria population, [math]q[/math], is modeled by the equation [math]q=500,000\cdot\left(\frac{1}{3}\right)^d[/math], where [math]d[/math] is the number of days since it was first measured.[br][/size][br]Which statement is true about the two populations?