Expert: Josh - 9/14/2004

Question
I need to formulate three word problems that can be translated into quadratic equations ... ugh! Can you help?  

Answer
Hi Sharon,

The possibilties are endless.

First of all, understand that a quadratic equation has the form, a*x^2+b*x+c=0, where the coefficients a,b and c represent some constants (numbers that you know as you derive the equation). The only unknown quantity is "x".
"x" can represent anything, it all depends on the context of the problem.

Here are the conditions that you must abide by.
-In your formulation, the degree of the polynomial (i.e., the equation in terms of x) must be no larger than 2.
-You must clearly define what "x" represents.
-You should indicate what to solve for (explain the underlying problem, stating what your objective is).

Here is an example:

Daniel is a landscape gardener. He is building a fence for a rectangular enclosure which is to be used as a child play area. Due to budget constraints, he can only afford 80 meters of fencing, what should be the dimensions of the enclosure, if he has to maximize its area.

Solution:
A--------B



D--------C

Let the length |AB|=|DC|=x.   ...[#A]
Let the width |AD|=|BC|=y.    ...[#B]

The total area is A=xy. ...[#1]
But, the perimeter, P, must satisfy the constraint,
P=|AB|+|BC|+|CD|+|DA|=80.
Using [#A] and [#B],
P=2(x+y)=80 ...[#2a]
So, y=x-40  ...[#2b]

Substituting [#2b] into [#1], we get
A=x(x-40)
A=x^2-40x.

Have you done differentiation before?
To maximize the area (A), we need to take the first derivative of A (denoted dA/dx or A') and set it to zero. What this is doing is we are attempting to find the stationary point A(x) on the up-side-down parabola, where the area is maximized.
The quadratic equation itself looks like an inverted "U" when you draw it on a graph.

Solving for dA/dx = 2x-40 = 0, x=20.
So, it turns out that the area will be maximized by setting x=20, y=40-x=20. [If in doubt, plot the quadratic equation on a piece of grid paper to understand this more thoroughly.]

What this means in general, is that to maximize the area of a rectangular enclosure, when you have a limited amount of material is to build a square enclosure.
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Any meaningful problem that involves quadratic formulation amounts to solving a quadratic equation, one way or another. Inevitably, it involves differentiation, because in most physical problems, we are interested in maximizing or minimizing the objective function (as we have seen in the example above).

Why don't you talk it over with friends? and let your creativity and imagination get to work.

Cheers,
Josh