On remembering why I love Math and Completing the Square

I just finished function theory and went straight to polynomials. It was refreshing to meet and work with these old friends of mine. And unlike a week ago, I was a little readier to meet this:
ax² + bx + c = 0
 

On January 5th, I wrote about my rude awakening when I realized I remember a lot less math, that I thought. The example I wrote about was a transformation of ax² + bx + c = 0 to

 
Well, guess what?, not only can I now do the transformation myself and understand its significance, but the journey to learning it made me remember why I loved math. I'll show you the process I penciled in.

I would be lying if I said this progress did not thrill me. I am experiencing the same excitement I used to get in high school when things started to click. In the grand scheme of things, the transformation I just did is trivial. However, within the process, I was reminded to do a step that, for me, it is the magic of mathematics. I got to ad a number to both sides of the equation that was not there before.
 
Yes, indeed. In ordert to go from ax² + bx + c = 0 to I had to use a technique called, completing the square. That technique allows us to remove the exponent of x² in order to have only x variables in the equation.
Therefore if I wanted to express ax² + bx + c = 0 in terms of x I would need to do the following.

First I need to get rid of the coefficient a in ax² that is done by the whole polynomial dividing by a:
x² + ^b/_ax + ^c/_a = 0.
 
Then we pass the non-variable term to the other side of the equation.
x² + ^b/_ax = - ^c/_a
 
Now here is where the magic happens. When we study quadratic formulas we learn that x² +2bx +b² is a perfect square that can be expressed as (x + b) ². So in theory whenever we get an equation that fits the mold x² + 2bx, we can ad +b² (which would mean half of whatever number b is and then squaring that number) on both sides of the equation and complete the square. I will do that to our main equation bellow.
 
 
x² + ^b/_ax + (^b/_2a)² = - ^c/_a + (^b/_2a)²


Now we can get


(x + ^b/_2a)² = (^b/_2a)² - ^c/_a

And after simplifying we get.

(x + ^b/_2a)² = ^b2/_4a² - ^c/_a


 
 
Then applying another technique I love, we take ^c/_a and transform it into something we can subtract from ^b2/_4a². All we need to do is multiply it by ^4a/_4a , the equivalent of multiplying by one, thus leaving the term unchanged.

Now we have

(x + ^b/_2a)² = ^b2/_4a² - ^4ac/_4a²
 
Which finally simplifies to
 
(x + ^b/_2a)² = ^{b2 - 4ac}/_4a²
 
 
And there it is. What caused me so much grief a week ago, now is understood. But more importantly, I rediscovered two powerful tools in mathematics.
 
1) All thigs being equal...we can add a term or operation to both side of an = sign and not change it.
So if x = c, the x +1 = c +1; and (x+1)² = (c+1)²
 
2) Identity
Math is a language. It describes our world. Therefore it should not be surprising that it does things we do in real life.
For example, number 1 is both 1 and 1/1, and 100/100, and 1² , and 4-3, etc. They are all 1. However, if I am dealing with money knowing that I can express 1 dollar as 100/100 cents is really useful.
 
The same things happens with people. If you have a brother, he is also your mother's (or father's) son, he is your grandmother's grandson, your uncle's nephew, etc. In some contexts it would be more important for someone to know your brother as your grandmother's grandson, than as your uncle's nephew.
 
Isn't math great? Let me know what you think in the comments.