In last weeks problem we had a hypothetical scenario of two quarterbacks, called “Brady”: and “Manning”: Brady threw the football given by the quadratic equation:

y = -x² + 180x – 7700

Manning threw the football given by the equation:

y = -x² + 140x – 4500

Y represented the height of the football above the ground in yards.
X represented the yards on the football field.

Since each player is about 6 feet tall, we will try to solve the equation for x when y = 2, not zero. Why 2? Remember y is in yards, 2 yards is roughly 6 ft. (We could use the zeroes to solve the problem if we wanted to be a little less accurate)

So our new equations are:

2 = -x² + 180x – 7700 (Brady)
2 = -x² + 140x – 4500 (Manning)

So we have three potential methods of solving the equation: the Quadratic Formula, Factoring, and Completing the Square. Since these equations may not even be factorable and it may be difficult to complete the square in this instance I recommend using the Quadratic Formula.

To refresh your memory the Quadratic Formula is:
x = -b +_ √ b² – 4ac 

This is of course all divided by 2a.

So now we have to convert our equations to the form 0 = ax² + bx + c

This brings our equations to:
0 = -x² + 180x – 7702 (Brady)
0 = -x² + 140x – 4502 (Manning)

So in this case for Brady a = -1, b = 180, and c = -7702
For Manning a = -1, b = 140, and c = – 4502

O.k., now plug those in …

Did you get x values of 70.05 and 109.95 for Brady? I did. That means Brady threw from slightly past the 70 yard line and his receiver caught it in the end zone. Touchdown! Oh yeah!

Now plug it in for Manning …

Did you get x values of 50.05 and 89.95 for Manning? I did. That means Manning threw it from slightly past the 50 yard line and his receiver caught it slightly before the 90 yard line. No touchdown for Manning.

So which one threw it farther? To figure that out use simple subtraction:

Brady: 109.95 – 70.05 = 39.9 yards
Manning: 89.95 – 50.05 = 39.9 yards

So both Brady and Manning threw the same distance. From this I would conclude that Brady simply had a better team that got him further up the field.

O.k. so is this problem realistic? Yes, I know that the Broncos won the Superbowl, but I assume the Pats will beat them next year. What I really want to look at is the equations. I want to see the max height of the football in the air. To do this I have to find the axis of symmetry or the vertical line that goes through the vertex of the parabola. The vertex is where the parabola, or in this case the football, reaches maximum height.

The formula for the axis of symmetry is x = -b/2a. Figuring that out for Brady the axis of symmetry is x = 90, for Manning x = 70. Plugging those values into our original equations we will get:

y = -(90)² + 180(90) – 7700 = 400 yards in height for Brady and
y = -(70)² + 140(70) – 4500 = 400 yards in height for Manning as well.

So would both football players throw it a height equal to 4 football fields? I don’t think so. Stay tuned next week for how we fix these equations …

If anything has confused you with this explanation remember that I am here to help. Feel free to contact me to arrange private tutoring.

For Teachers: Feel free to use this problem in your Algebra or Integrated Math 2 class.

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