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Comparing Methods for Solving Quadratics

Compare the Quadratic Formula, factoring, and taking the root of both sides.

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Real World Applications – Algebra I


When’s the best time to blow up fireworks?

Student Exploration

The Naperville Central freshman football team has just won their DVC tournament. To celebrate the school is putting on a fireworks display and the team is helping with the planning.

The fireworks will use rockets launched from the top the tower near the school which is 160 feet off the ground. The mechanism will launch the rockets so that they are initially rising 96 feet per second.

The team members want the fireworks from each rocket to explode when the rocket is at the top of its trajectory. They need to know how long it will take the rocket to reach the top so that they can set the timing mechanism. The team members would also like to inform spectators of the best place to stand and see the fireworks so that they need to know how high the rocket will go.

Ricky is on the varsity football team and a member of the National Honor Society and has volunteered to help plan the fireworks display. Because he is in AP Physics, Ricky knows that there is a function \begin{align*}h(t)\end{align*} that will give the rocket's height off the ground in terms of t, time elapsed since the launch: \begin{align*}h(t) = 160 + 96t - 16t^2\end{align*}.

Your task is to help the football team in planning the fireworks display.

  1. Draw a sketch of the situation.
  2. Write a clear statement of the questions the football team wants answered.
  3. Describe what you must do to Ricky’s function to complete question #4.
  4. Consider a change in the tower height and how this would affect the height of the fireworks. Let the tower height be 100ft versus 160ft. Record your results. Let the tower height be 200ft, and record your results. Write a clear statement of the questions the football team wants answered for each height.
  5. Analyze how the tower height affected the firework height and draw a conclusion. Hypothesize what might happen if the tower was 300ft or 500ft. Write your conclusion and justify your answer.

Extension Investigation


  1. Draw a sketch of the situation.
  2. What do each of the numbers in the equation represent in the situation?
  3. If the fireworks don’t explode, how long would it take for the rocket to reach the ground?
    1. Set \begin{align*}h(t) = 0\end{align*}, and solve this using the quadratic formula. What does your answer mean in this situation?
    2. Set \begin{align*}h(t) = 0\end{align*}, and solve this by completing the square.
      1. What is the first step to complete the square? Why?
      2. Continue to write and explain each step to complete the square.
      3. What does your solution have to do with the situation?
      4. What is the vertex? What do these numbers mean?
    3. Compare your answers for a & b and your sketch. Did you get the same answers? Why or why not?
    4. Do your answers make sense? Why or why not?
    5. When would the fireworks be at a height of 20ft? Use any method to solve. Do your answers make sense? Why or why not?
  4. If the height of the tower was 100ft instead of 160ft, how does this change the rocket’s path? (You might want to find different ways of representing this situation!) What if the height of the tower were 200ft? 300ft?
    1. What patterns or conclusions can you draw from the different heights of the tower?
  5. When solving quadratic equations, which method is the easiest? Why? Which method is the most challenging? Why?


6. Is there a way that this equation can be rewritten in vertex form?
a. If this is possible, try solving this quadratic using square roots. What similarities and differences do you notice between this method of solving and the other two methods?

Resources Cited

Original Activity found here: http://www.google.com/url?sa=t&rct=j&q=&esrc=s&source=web&cd=4&ved=0CFYQFjAD&url=http%3A%2F%2Fcmarzock.wikispaces.com%2Ffile%2Fview%2FActivity%2BQuadratic%2BFireworks.docx&ei=5hDyT5tghfjaBY79ucEK&usg=AFQjCNHxZgdC6q68koYsIqjygBM-hCQVXg

Connections to other CK-12 Subject Areas


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