Consider the quadratic equation x2 - 6x - 16 = 0. a. solve the equation, and discuss how the type of roots affects the appearance of the graph of the function y = x2 - 6x - 16. b. what is the vertex of the graph of x2 - 6x - 16 = 0? c. what is the equation of the axis of symmetry? d. use this information to draw a rough sketch of the graph of the function.

The area is given by A = -2x² + 120x. The greatest area is given by x = 30 m.

Explanation:
See the picture attached for reference.

Let's call:
x = side not facing the wall
We know that the total fence is 120m, therefore
(120 - 2x) = side facing the wall

Note: you could choose to be x = side facing the wall, but the calculations would be a little bit more complicated.

We can now calculate  the area of a rectangle:
A = b · h =
   = x · (120 - 2x)
   = -2x² + 120x

In order to find a maximum for this function, we need to calculate the first derivative:


Then, we need to find the candidate points by setting the derivative equal to zero:
-4x +120 = 0
-4(x - 30) = 0
x = 30

Now, in order to understand if the candidate point is a maximum or a minium, let's calculate the second derivative:



According to the "Second Derivative Test", if the second derivative is negative, the point is a local maximum.

Hence, x = 30m gives the greatest area, and we would have:
side not facing the wall = 30m
side facing the wall = 120 - 2·30 = 60m
area = 30 · 60 = 1800 m²