In English it is taught to be more complex, more descriptive. But as we all know, a good book leaves a lot of details to the imagination. Math is the opposite of English. Here the unknown is not to be made complex, but it is to be described in simple, yet quite powerful, definitions. The simple is used to explain the complex.
Using college algebra and some calculus to calculate the slopes of a graph, we can compare a known parabola’s properties and values to an unknown parabola, thus creating a simple alternative to the Quadratic Equation.
Given: Given the values of a known parabola. (Example: A simple one such as x^2.) And equations of the known and unknown in the form of “ax^2 + bx + c” are also known. The cumbersome equations and often occurrence of no real value of the Quadratic Equation can be avoided by a simple observation of how parabolas change and how those changes relate to one another.
We also know that special involutes give a value of a circle with changing radius. Also we note that special parabolas’ relative coordinates also show a circle of changing radius. (See another Math Hunches.)
Here is a brief description of the process:
Take the slope between the parabola of y=x^2 (a common parabola were the values are known) and the unknown parabola. Do this by finding the x-intercepts of both “y= ax^2 + bx + c” equations. (Recall these equations are this given. Substitute zero for x.)
Each time the absolute coordinates (x-plane) of x^2 changes by a distance of pi radians the unknown value changes proportionally. This proportion is found by finding the slope between “y= x^2", x-intercept and the unknown parabola’s x-intercept. This is essentially finding where the parabolas match up. It is as if both started at (0,0).
Similarly: It is possibly to use an involute to describe the distance between the y= x^2 value and the unknown value. Take the y coordinate difference between x^2 and the unknown. For every increment of slope of the slope between the x-intercepts; the absolute value of the unknown parabola’s coordinates equals “x^2 + pi radians * y coordinate difference.”
Of course it is important to note, this is just a fast theory. It might not even work. However I have been working on this idea for a while and this is my best work so far. I will place it also at www.absolute-proof.net/wiki . That is a new Wiki that complements the work at Constructor’s Corner. If nothing else the theory is intriguing and can lead to better, more accurate ideas.