Three Fundamental Concepts Of Mathematics (Dealing With Equations)

Hi, I love mathematics and have learned / made up these concepts when doing math.

Hopefully after some example problems you get the gist of the three concepts I am trying to convey.

Example #1:

The equation Ax^2 + B*x + C = 0.

There are three general dependencies of equation which are A, B, and C.

Imagine that you wanted to solve this graphically, then that would mean you would have to make a different graph for different coefficients.

Let us reduce this to two dependencies by dividing by A and assuming A does not equal to 0.

x^2 + (B/A)x + (C/A) = 0

Subtract C/A from both sides.

x^2 + (B/A)x = -(C/A)

Now here is the 2nd concept which is I like to call "Identity Transformation".

(-C/A) = x^2 + (B/A)x = x^2 + 2(B/2A)x

Notice that we are "rewriting" B/A, because then we can do what is called completing the square.

x^2+2(B/2A)x + (B/2A)^2 = (B/2A)^2 - (C/A)

Time to use the "Identity Transformation" once more

(B/2A)^2 - (C/A) = x^2+2(B/2A)x + (B/2A)^2 = (x + B/2A)^2

Now, if you were to substitute x + B/2A as z and (B/2A)^2 - (C/A) as K, then you would get 

z^2 = K

Time to introduce the "Complexity Reduction" concept. Notice how z is raised to the power of 2, and never any other number. Because of this we will take the inverse of z^2 on the solution K which is +- sqrt(K). Comparing this to solving the original equation through graphing, we only need to graph y = x^2 and we never need to change it based on the constant coefficients. I like to call this "complexity reduction". Reducing the equation to having one side be a transformation of a variable with no general dependencies and the other side being a transformation of known dependencies both general and specific. Now you might be tempted to say well isn't the exponent 2 a dependency, well yes BUT it will always be 2 with respect to the original equation. So 2 is what I like to call a specific dependency while the constant coefficients are general dependencies cause you can change them (but you assume you know their value so they are not variables). So taking the inverse transformation is "allowed" because the transformation is not going to change if you change the coefficients of the original equation (it is always going to be x^2 and +-sqrt(x)). So you basically reduced the need to draw a different graph for different coefficients to just needing to draw one graph (if you were trying to solve it graphically)

So you take the inverse of x^2 on K and you get z = +- sqrt(K)

NOW you substitute z with x + B/2A

So how can you get all general dependencies of z to the right side, you subtract B/2A of course. and Wallah, you have the solution x.

*** To be continued **