Rewriting Quadratic Expressions (Part2): IM Alg1.7.7
[url=https://im.kendallhunt.com/HS/teachers/1/7/7/preparation.html]“Rewriting Quadratic Expressions in Factored Form (Part 2)”[/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
- Alg1.7.7 Lesson
- IM Alg1.7.7 Lesson: Rewriting Quadratic Expressions in Factored Form (Part 2)
- Alg1.7.7 Practice
- IM Alg1.7.7 Practice: Rewriting Quadratic Expressions in Factored Form (Part 2)
IM Alg1.7.7 Lesson: Rewriting Quadratic Expressions in Factored Form (Part 2)
The product of the integers 2 and -6 is -12.
List all the other pairs of integers whose product is -12.
Of the pairs of factors you found, list all pairs that have a positive sum. Explain why they all have a positive sum.
Of the pairs of factors you found, list all pairs that have a negative sum. Explain why they all have a negative sum.
These expressions are like the ones we have seen before. Each row has a pair of equivalent expressions. Complete the table. If you get stuck, consider drawing a diagram in the applet below.
These expressions are in some ways unlike the ones we have seen before. Each row has a pair of equivalent expressions. Complete the table. If you get stuck, consider drawing a diagram in the applet above.
Name some ways that the expressions in the second table are different from those in the first table (aside from the fact that the expressions use different numbers).
[size=150]Consider the expression [math]x^2+bx+100[/math].[br][br]Complete the first table with all pairs of factors of 100 that would give positive values of [math]b[/math], and the second table with factors that would give negative values of [math]b[/math].[/size][br][br]For each pair, state the [math]b[/math] value they produce. (Use as many rows as needed.)
[size=150]Consider the expression [math]x^2+bx-100[/math]. Complete the first table with all pairs of factors of -100 that would result in positive values of [math]b[/math], the second table with factors that would result in negative values of [math]b[/math], and the third table with factors that would result in a zero value of [math]b[/math].[/size][br][br]For each pair of factors, state the [math]b[/math] value they produce. (Use as many rows as there are pairs of factors. You may not need all the rows.)
Write each expression in factored form:
[math]x^2-25x+100[/math]
[math]x^2+15x-100[/math]
[math]x^2-15x-100[/math]
[math]x^2+99x-100[/math]
How many different integers [math]b[/math] can you find so that the expression [math]x^2+10x+b[/math] can be written in factored form?
IM Alg1.7.7 Practice: Rewriting Quadratic Expressions in Factored Form (Part 2)
Find two numbers that...
multiply to -40 and add to -6.[br][br]If you get stuck, try listing all the factors of the first number.
multiply to -40 and add to 6.
multiply to -36 and add to 9.
multiply to -36 and add to -5.
Create a diagram to show that (x-5)(x+8) is equivalent to x²+3x+40.
Write a + or a - sign in each box so the expressions on each side of the equal sign are equivalent.
Match each quadratic expression in standard form with its equivalent expression in factored form.
Rewrite each expression in factored form. If you get stuck, try drawing a diagram in the applet below.
[math]x^2-3x-28[/math]
[math]x^2+3x-28[/math]
[math]x^2+12x-28[/math]
[math]x^2-28x-60[/math]
[size=150]Which equation has exactly one solution?[/size]
The graph represents the height of a passenger car on a ferris wheel, in feet, as a function of time, in seconds.
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[/img][br][size=150]Use the graph to help you:[/size][br][br]Find [math]H(0)[/math].
Does [math]H(t)=0[/math] have a solution? Explain how you know.
Describe the domain of the function.[br][br]
Describe the range of the function.
[size=150]Elena solves the equation [math]x^2=7x[/math] by dividing both sides by x to get [math]x=7[/math]. She says the solution is 7.[br][br]Lin solves the equation [math]x^2=7x[/math] by rewriting the equation to get [math]x^2-7x=0[/math]. When she graphs the equation [math]y=x^2-7x[/math], the [math]x[/math]-intercepts are [math](0,0)[/math] and [math](7,0)[/math]. She says the solutions are 0 and 7.[/size][br][br]Do you agree with either of them? Explain or show how you know.
[size=150]A bacteria population, [math]p[/math], can be represented by the equation [math]p=100,000\cdot\left(\frac{1}{4}\right)^d[/math], where [math]d[/math] is the number of days since it was measured.[/size][br][br]What was the population 3 days before it was measured? Explain how you know.
What is the last day when the population was more than 1,000,000? Explain how you know.[br]