Applying the Quadratic Formula (Part 1): IM Alg1.7.17
[url=https://im.kendallhunt.com/HS/teachers/1/7/17/preparation.html]“Applying the Quadratic Formula (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
- Alg1.7.17 Lesson
- IM Alg1.7.17 Lesson: Applying the Quadratic Formula (Part 1)
- Alg1.7.17 Practice
- IM Alg1.7.17 Practice: Applying the Quadratic Formula (Part 1)
IM Alg1.7.17 Lesson: Applying the Quadratic Formula (Part 1)
Here is an example of someone solving a quadratic equation that has no solutions:
[center][math]\displaystyle \begin {align} (x+3)^2+9 &=0\\ (x+3)^2 &=\text-9\\ x+3 &=\pm \sqrt{\text-9} \end {align}[/math][/center]Study the example. At what point did you realize the equation had no solutions?
Explain how you know the equation [math]49+x^2=0[/math] has no solutions.[br]
Answer each question without graphing. Explain or show your reasoning.
[size=150]The equation [math]h\left(t\right)=-16t^2+80t+64[/math] represented the height, in feet, of a potato [math]t[/math] seconds after it has been launched.[/size][br][br][size=100]Write an equation that can be solved to find when the potato hits the ground. Then solve the equation.[br][/size]
Write an equation that can be solved to find when the potato is 40 feet off the ground. Then solve the equation.[br]
[size=150]The equation [math]g\left(t\right)=2+23.7t-4.9t^2[/math] models the height, in meters, of a pumpkin [math]t[/math] seconds after it has been launched from a catapult.[/size][br][br]I[size=100]s the pumpkin still in the air 8 seconds later? Explain or show how you know.[/size][br]
At what value of [math]t[/math] does the pumpkin hit the ground? Show your reasoning.[br]
In an earlier lesson, we tried to frame a picture that was 7 inches by 4 inches using an entire sheet of paper that was 4 inches by 2.5 inches.
[size=150]One equation we wrote was [math]\left(7+2x\right)\left(4+2x\right)=38[/math].[/size]
Explain or show what the equation [math]\left(7+2x\right)\left(4+2x\right)=38[/math] tells us about the situation and what it would mean to solve it. Use the diagram, as needed.
Solve the equation without graphing. Show your reasoning.[br]
[size=150]Suppose you have another picture that is 10 inches by 5 inches, and are now using a fancy paper that is 8.5 inches by 4 inches to frame the picture. Again, the frame is to be uniform in thickness all the way [br]around. No fancy framing paper is to be wasted![/size][br][br]Find out how thick the frame should be.
[size=150]Suppose that your border paper is 6 inches by 8 inches. You want to use all the paper to make a half-inch border around some rectangular picture.[br][/size][br]Find two possible pairs of length and width of a rectangular picture that could be framed with a half-inch border and no leftover materials. [br]
What must be true about the length and width of any rectangular picture that can be framed this way? Explain how you know.[br]
IM Alg1.7.17 Practice: Applying the Quadratic Formula (Part 1)
[size=150]Select [b]all[/b] the equations that have 2 solutions.[/size]
[size=150]A frog jumps in the air. The height, in inches, of the frog is modeled by the function [math]h\left(t\right)=60t-75t^2[/math], where [math]t[/math] is the time after it jumped, measured in seconds.[/size][br][br]Solve [math]60t-75t^2=0[/math]. What do the solutions tell us about the jumping frog?
[size=150]A tennis ball is hit straight up in the air, and its height, in feet above the ground, is modeled by the equation [math]f\left(t\right)=4+12t-16t^2[/math], where [math]t[/math] is measured in seconds since the ball was thrown.[/size][br][br]Find the solutions to the equation [math]0=4+12t-16t^2[/math].
What do the solutions tell us about the tennis ball?[br]
Rewrite each quadratic expression in standard form.
[math]\left(x+1\right)\left(7x+2\right)[/math]
[math]\left(8x+1\right)\left(x-5\right)[/math]
[math]\left(2x+1\right)\left(2x-1\right)[/math]
[math]\left(4+x\right)\left(3x-2\right)[/math]
Find the missing expression in parentheses so that each pair of quadratic expressions is equivalent. Show that your expression meets this requirement.
[math](4x-1)(\underline{\hspace{1in}})[/math] and [math]16x^2-8x+1[/math]
[math](9x + 2)(\underline{\hspace{1in}})[/math] and [math]9x^2-16x-4[/math]
[math](\underline{\hspace{1in}})(\text{-}x + 5)[/math] and [math]-7x^2+36x-5[/math]
[size=150]The number of downloads of a song is a function, [math]f[/math], of the number of weeks, [math]w[/math], since the song was released. The equation [math]f\left(w\right)=100,000\cdot\left(\frac{9}{10}\right)^w[/math] defines this function.[br][/size][br]What does the number 100,000 tell you about the downloads?
What about the [math]\frac{9}{10}[/math]?
Is [math]f\left(-1\right)[/math] meaningful in this situation? Explain your reasoning.[br]
[size=100][size=150]Consider the equation [math]4x^2-4x-15=0[/math].[/size][br][br]I[/size][size=100]dentify the values of [math]a[/math], [math]b[/math], and [math]c[/math] that you would substitute into the quadratic formula to solve the equation.[/size][br]
Evaluate each expression using the values of a, b, and c.
[size=100]The solutions to the equation are [math]x=-\frac{3}{2}[/math] and [math]x=\frac{5}{2}[/math]. [/size]Do these match the values of the last expression you evaluated in the previous question?
[size=150]Describe the graph of [math]y=-x^2[/math]. [/size][br][br](Does it open upward or downward? Where is its [math]y[/math]-intercept? What about its [math]x[/math]-intercepts?)[br]
[size=150]Without graphing, describe how adding [math]16x[/math] to [math]-x^2[/math] would change each feature of the graph of [math]y=-x^2[/math]. (If you get stuck, consider writing the expression in factored form.)[br][/size][br]the [math]x[/math]-intercepts?
the vertex?
the [math]y[/math]-intercept?
the direction of opening of the U-shape graph?