1. How the U.S. Army Uses
Quadratic Equations

2. Soldiers in the U.S. Army use quadratic equations on a daily basis…

3. …especially if they work with missiles.

4. Cruise missile fired from a submarine:

5. 1. Describe the path of a missile through the air.
Curved up
(from launch)
down
(to target)

6. A curved path is something you have seen before:

7. 2. When you graph a quadratic equation, what is the curved
shape called?
A parabola

8. So, we can say that missiles follow a parabolic path, from launch site to target.

9. 3. Explain why the path of a missile is parabolic. What force
pushes it up? What force pulls it down?
Pushed up by the force of an explosion
Pulled down by the force of gravity
Launch point
Target

10. Since the missile follows a parabolic path, we can use a quadratic equation to describe its motion.
ax² + bx + c = y 2 3 5 4 1 6 6 4 5 3 1 2 3 1 1 2 2
height
Time
Should I write the equation with h and t?

11. 4. Sketch a parabola starting at point (0,0), passing through point (10, 10) and ending at point (0,20),

12. This parabola represents the flight path of a missile.

13. Label the X and Y axes.
Height of missile
(meters)
Time
(seconds)

14. Label the vertex.
vertex
Height of missile
(meters)
Time (seconds)

15. 5. What does the vertex tell us about this missile?
That it will reach a maximum height of 10 meters
Vertex
Height of missile
(meters)
Time (seconds)

16. 6. When will the missile reach the vertex? 10 seconds after launch
Height of missile
(meters)
Time (seconds)

17. 7. What happens at point (0, 20) on the graph?
The missile will hit the target/explode
Vertex
Height of missile
(meters)
Time (seconds)

18. 8. When will the missile hit the target? 20s after launch
Vertex
Height of missile
(meters)
Time (seconds)

19. 9. Your commanding officer tells you that a missile will be fired from 3m above the ground, with an upward velocity of 14 m/s. He wants to know when the missile will hit the target.

20. Downward acceleration due to gravity
Step 1: turn the problem into a quadratic equation.
ax²
Downward acceleration due to gravity bx
Upward velocity
14x 3 c
Starting position
Why is it written as 14x? Is x time (the unknown)? y
Height of missile

21. Downward acceleration due to gravity
ax²
Downward acceleration due to gravity
Negative bx
Upward velocity
14x 3 c
Starting position y
Height of missile

22. Gravity pulls all objects down at the same acceleration
Gravity pulls all objects down at the same acceleration. It doesn’t matter if the object is a feather, a bowling ball, or a cruise missile; gravity changes the position of an object by about 5 meters per second squared.
(Why ½ 10m/s²?) Is “rate the correct word?

23. Downward acceleration due to gravity
ax²
Downward acceleration due to gravity
-5x²
Negative bx
Upward velocity
14x 3 c
Starting position y
Height of missile

24. Downward acceleration due to gravity
What will the height of the missile be when it hits the target?
ax²
Downward acceleration due to gravity
-5x²
Negative bx
Upward velocity
14x 3 c
Starting position y
Height of missile

25. Downward acceleration due to gravity
Step 2: write the quadratic equation in standard form.
-5x² + 14x + 3
= 0
ax²
Downward acceleration due to gravity
-5x²
Negative bx
Upward velocity
14x 3 c
Starting position y
Height of missile

26. Step 3: factor the equation and solve for x.
First of all, how could we make this equation easier to solve?
-1 ( )
-5x² + 14x + 3
= 0
5x² - 14x - 3
= 0
(-15, 1)
-15
Re-write the middle of the equation using -15 and 1:
Does 15, -1 also work?
5x² - 15x + x - 3
= 0
Factor and solve:
5x (x – 3) + 1 (x – 3)
= 0
(5x + 1) (x – 3)
= 0
(5x + 1)
= 0 or
(x – 3)
= 0
x = or x = 3

27. 10. Remember, we are trying to figure out when the missile will hit the target. Of the two answers, which one makes sense, and why?
x= 3 is correct
The missile will hit the target 3 seconds after launch.
x = or x = 3
Time cannot be negative

28. 11. Your commanding officer tells you that a missile will be fired from 2m above the ground, with an upward velocity of 9m/s. He wants to know when the missile will hit the target.

29. Downward acceleration due to gravity
Step 1: turn the problem into a quadratic equation.
-5x²
ax²
Downward acceleration due to gravity 9x bx
Upward velocity 2 c
Starting position y
Height of missile

30. Downward acceleration due to gravity
Step 2: write the equation in standard form.
-5x² + 9x + 2
= 0
-5x²
ax²
Downward acceleration due to gravity
Negative 9x bx
Upward velocity 2 c
Starting position y
Height of missile

31. Step 3: factor the equation and solve for x.
-1 ( )
-5x² + 9x + 2
= 0
5x² - 9x - 2
= 0
The missile will hit the target 2 seconds after launch.
5x² - 10x + x -2
= 0
5x (x – 2) + 1 (x – 2)
= 0
(5x + 1) (x – 2)
= 0
(5x + 1)
= 0 or
(x – 2)
= 0
x = or x = 2
Does 15, -1 also work?