Quadratic Equations are used to find maximums and minimums for rectangular regions. An example of this type of problem would occur when a person, with a specific amount of fencing, wants to find the largest rectangular

area that can be fenced off.

In order to solve quadratic equations involving maximums and minimums for rectangular regions, it is necessary to

Suppose a farmer has 1000 yards of fencing to enclose a rectangular field. What is the largest

area that the farmer can enclose?

The figure shown below illustrates the rectangular fence that is to be built. The fence will surround the rectangular area, and therefore, will create the

*perimeter* of the region.

The region inside the fence is described by area. An

expression for this figure's

area would be:

We need to solve the

perimeter formula for either

*l* or

*w*. Let’s solve for

*w*:

Now substituting

into the

area formula we have:

Since A represents a

quadratic equation (

) in terms of

*l*, we will re-write A in

function form with the exponents in descending order:

The

graph of

will be a

parabola and, since

, the

parabola will have a maximum

point as its vertex. The y-coordinate of the

vertex will represent our greatest area. To proceed, we need to find the value of the x-coordinate of the

vertex (that is, the value of

*l* in our equation).

Substituting this value for

*l* into our

equation for

area yields:

Therefore the largest

area that the farmer could enclose would be a

square where each

side has a

length 250 yards.