Applying the Quadratic Formula (Part 1)

ID: gimab-takin
Created by Illustrative MathematicsIllustrative Mathematics, CC BY 4.0
Subject: Algebra, Algebra 2
Grade: 8-9
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14 questions

Applying the Quadratic Formula (Part 1)

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No Solutions for You!

Here is an example of someone solving a quadratic equation that has no solutions:

(x+3)2+9=0(x+3)2=9x+3=±9\begin{array}{rcl} \\[-1em] (x + 3)^{2} + 9 &=&0 \\[-1em] \\ \\[-1em] (x + 3)^{2} &=& -9 \\[-1em] \\ \\[-1em] x + 3 &=& \pm \sqrt{-9 }\\[-1em] \end{array}

1) Study the example. At what point did you realize the equation had no solutions?

2) Explain how you know the equation 49+x2=049 + x^{2} = 0 has no solutions.

The Potato and the Pumpkin

Answer each question without graphing. Explain or show your reasoning.

The equation h(t)=16t2+80t+64h(t) = -16t^{2} + 80t + 64 represented the height, in feet, of a potato tt seconds after it has been launched.

3) Write an equation that can be solved to find when the potato hits the ground.

4) Then solve the equation.

Show Work

5) Write an equation that can be solved to find when the potato is 40 feet off the ground.

6) Then solve the equation.

Show Work

The equation g(t)=2+23.7t4.9t2g(t) = 2 + 23.7t - 4.9t^{2} models the height, in meters, of a pumpkin tt seconds after it has been launched from a catapult.

7) Is the pumpkin still in the air 8 seconds later?

True or false? Write below.

8) Explain or show how you know.

9) At what value of tt does the pumpkin hit the ground?

10) Show your reasoning.

Back to the Framer

Suppose you 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. One equation you could write to help you could be (7+2x)(4+2x)=38(7+ 2x)(4 + 2x) = 38.

11) Explain or show what the equation (7+2x)(4+2x)=38(7+ 2x)(4 + 2x) = 38 tells us about the situation and what it would mean to solve it. Use the diagram, as needed.

A template for answering this question. Ask your instructor for an alternative.

12) Solve the equation without graphing.

13) Show your reasoning.

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 around. No fancy framing paper is to be wasted!

14) Find out how thick the frame should be.

Show Work