Rewriting Quadratic Expressions (Part1): IM Alg1.7.6
[url=https://im.kendallhunt.com/HS/teachers/1/7/6/preparation.html]“Rewriting Quadratic Expressions in Factored Form (Part 1)”[/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
- Alg1.7.6 Lesson
- IM Alg1.7.6 Lesson: Rewriting Quadratic Expressions in Factored Form (Part 1)
- Alg1.7.6 Practice
- IM Alg1.7.6 Practice: Rewriting Quadratic Expressions in Factored Form (Part 1)
IM Alg1.7.6 Lesson: Rewriting Quadratic Expressions in Factored Form (Part 1)
Here are puzzles that involve side lengths and areas of rectangles. Can you find the missing area in Figure A? Be prepared to explain your reasoning.
Can you find the missing length in Figure B? Be prepared to explain your reasoning.
Use a diagram to show that each pair of expressions is equivalent.
[size=100]Observe the pairs of expressions that involve the product of two sums or two differences. [/size][br][br]How is each expression in factored form related to the equivalent expression in standard form?[br]
Each row in the table contains a pair of equivalent expressions. Complete the table with the missing expressions. If you get stuck, consider drawing a diagram.
[size=150]A mathematician threw a party. She told her guests, “I have a riddle for you. I have three daughters. The product of their ages is 72. The sum of their ages is the same as my house number. How old are my daughters?”[br][br]The guests went outside to look at the house number. They thought for a few minutes, and then said, “This riddle can’t be solved!”[br][br]The mathematician said, “Oh yes, I forgot to tell you the last clue. My youngest daughter prefers strawberry ice cream.”[/size][br][br]With this last clue, the guests could solve the riddle. How old are the mathematician’s daughters?
IM Alg1.7.6 Practice: Rewriting Quadratic Expressions in Factored Form (Part 1)
Find two numbers that satisfy the requirements. If you get stuck, try listing all the factors of the first number.
Find two numbers that multiply to 17 and add to 18.
Find two numbers that multiply to 20 and add to 9.
Find two numbers that multiply to 11 and add to -12.
Find two numbers that multiply to 36 and add to -20.
Use the diagram to show that (x+4)(x+2) is equivalent to x²+6x+8.
Use the diagram to show that (x-10)(x-3) is equivalent to x²-13x+30.
[center][/center]Select [b]all[/b] expressions that are equivalent to [math]x-5[/math].
Here are pairs of equivalent expressions—one in standard form and the other in factored form. Find the missing numbers.
[math]x^2 +\boxed{\quad}x + \boxed{\quad}[/math] and [math](x-9)(x-3)[/math]
[math]x^2+12x+32[/math] and [math](x+4)\left(x+\boxed{\quad}\right)[/math]
[math]x^2-12x+35[/math] and [math](x-5)\left(x+\boxed{\quad}\right)[/math]
[math]x^2-9x+20[/math] and [math](x-4)\left(x+\boxed{\quad}\right)[/math]
Find all the values for the variable that make each equation true.
[math]b(b-4.5)=0[/math]
[math](7x+14)(7x+14)=0[/math]
[math](2x+4)(x-4)=0[/math]
[math](\text{-}2+u)(3-u)=0[/math]
[size=150]Lin charges $5.50 per hour to babysit. The amount of money earned, in dollars, is a function of the number of hours that she babysits.[/size][br][br]Which of the following inputs is impossible for this function?
[size=150]Consider the function [math]p(x)=\frac{x-3}{2x-6}[/math].[br][br][/size]Evaluate [math]p(1)[/math], writing out every step.[br]
Evaluate [math]p(3)[/math], writing out every step. You will run into some trouble. Describe it.
What is a possible domain for [math]p[/math]?
[size=150]When solving the equation [math](2-x)(x+1)=11[/math], Priya graphs [math]y=(2-x)(x+1)-11[/math] and then looks to find where the graph crosses the [math]x[/math]-axis.[br][br]Tyler looks at her work and says that graphing is unnecessary and Priya can set up the equations [math]2-x=11[/math] and [math]x+1=11[/math], so the solutions are [math]x=\text{-}9[/math] or [math]x=10[/math].[br][/size][br]Do you agree with Tyler? If not, where is the mistake in his reasoning?[br]
How many solutions does the equation have? Find out by graphing Priya’s equation.