Appendix 1
The well-known quadratic formula gives an example of the all-encompassing power of parameters. Examples of quadratic equations include the following:
1. 6x^{2} + 13x + 8 = 0.
2. x^{2} + 11x + 5 = 0.
3. 20x^{2} + 5x – 36 = 0.
4. x^{2} – 7x + 72 = 0.
Techniques for solving some problems of these types existed going all the way back to ancient Babylon over 3500 years ago—although it was only in the sixteenth and seventeenth centuries that the presentation of these equations would be similar in form to the way that the four examples are written.
One of the general methods for solving these equations is known as “completing the square.” The method can be applied to each of these equations to find individual solution(s) for each.
In the language of Chapter 4, we can think of each quadratic equation as describing an individual scenario where the three fundamental types of behavior (x^{2} behavior, x behavior, and constant behavior) each have a specific numerical value (or “price in dollars”) assigned to them. The values of these “prices” can change from scenario to scenario but remain constant in a given scenario and are what we want to capture as parameters. Note that these “price values” are often called “numerical coefficients” or “given values” or “givens” for a specific scenario.
So, in Equation 1 from the list, the price value for x^{2} is 6, for x is 13, and for the constant is 8. Once these price values are fixed, the x is still allowed to take on different values (like it was in the break-even situation from Chapter 4, where it represented the number of meals in a specific scenario).
For Equation 2, the price this time for x^{2} is 1, for x is 11, and for the constant is 5. And we get a new scenario, where the x is still free to take on different values. Similar situations occur for Equations 3 and 4.
Just as we were able to use P, C, and F to represent the dollar values of selling price per item, cost to make each item, and fixed costs, respectively, in the break-even scenario, we can use letters here to represent the “price values” for the three terms x^{2}, x, and the constant. The standard letters to use here are those early in the alphabet: a, b, and c, respectively.
Doing so, we obtain the following equation: ax^{2} + bx + c = 0. The three letters a, b, and c represent parameters, and what they give us is the power to represent all quadratic equations by a single super-equation. All of the individual equations can be obtained from this one by appropriate choices of a, b, and c. We illustrate this in the following table:
Set Parameters |
|||
a |
b |
c |
Super Equation ax^{2} + bx + c = 0 Becomes: |
6 |
13 |
8 |
6x^{2} + 13x + 8 = 0 |
1 |
11 |
5 |
x^{2} + 11x + 5 = 0 |
20 |
5 |
–36 |
20x^{2} + 5x – 36 = 0 |
–7 |
72 |
x^{2} – 7x + 72 = 0 |
Reasoning similarly, we can obtain all of the infinitely many other quadratic equations by appropriate choices of a, b, and c from the general equation.
We are not done, however.
The coup de grâce is that the method of completing the square, which can be used to solve each individual equation in its specific scenario, can now be applied to the super-equation (ax^{2} + bx + c = 0) to yield a general solution to the infinitely many equations all at once—in one grand maneuver. Doing so in this case ultimately yields the famous quadratic formula (details not shown):
The formula represents in writing a crystallization of the entire process of completing the square. It also means that once we identify the parameters in a given quadratic equation (which can be done on sight), instead of having to perform the more involved method of completing the square each time, we can simply plug the values for a, b, and c into the crystallized quadratic formula and do some arithmetic, and the solution pops out for us.
Moreover, this technique of using parameters to freeze-dry in writing the results of many algebraic maneuvers is not limited to the quadratic equation here or the break-even equation in Chapter 4, but can be used in all kinds of other situations with similar effect. This is what we have called big algebra.
Fertilizing the soil for others to purposefully use and systematically apply parameters for wide-scale impact and insight is—in the mind of many mathematical historians—Viète’s most important and revolutionary contribution to mathematics.