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**1.Homework Statement**

Derive the quadradic equation, [tex]ax^2 + bx + c = 0[/tex], using the fallowing method.

1. Divide by a, if a =/ 0.

2. Move the constant to the other side of the equation.

3. Square half the coeffecient of bx and add it to both sides, fixate the negative sign on -c/a.

4. factor the left side, simplify the right side.

5. apply the square root property.

6. solve for x.

## Homework Equations

[tex]ax^2+bx+c=0[/tex], where a is either 0 or not 0, if it is zero you skip the first step. (duh!)

[tex]x=\frac{-b\pm \sqrt{b^2-4ac}}{2a}[/tex]

## The Attempt at a Solution

1.[tex]ax^2+bx+c=0[/tex]

2.[tex]x^2+\frac{b}{a}x+\frac{c}{a}=0[/tex]

3.[tex]x^2+\frac{b}{a}x+=\frac{-c}{a}[/tex]

4.[tex]x^2+\frac{b}{a}x+\frac{1}{2}\left (\frac{b}{a} \right )^2=-\frac{c}{a}+\frac{1}{2}\left (\frac{b}{a} \right )^2 [/tex]

5.[tex]\left (x+\frac{b}{a} \right )^2=-\frac{c}{a}+\frac{1}{2}\left (\frac{b}{a} \right )^2 [/tex]

I'm going to split this step up because this is where I run into lots of problems of my understanding of deriving and deriving the quadratic formula.

So, I factored the left side, but can someone explain to me the logic and the why behind this? why is THIS correct? how does it WORK? I do not understand why it factors to [tex]\left (x+\frac{b}{a} \right )^2 [/tex]=[, and therefor I cannot attach it to my intuition, I'm just going off memorization of rules and rote, that an't the way I do things! I need to UNDERSTAND!

**EDIT 1: I do not know WHY! the last four tex coded equations aren't working! can someone help me fix that? it would help others in reading the problem I'm presenting, geeze... PLEASE bear with that until I figure out a way to get it fix, I've tried many things over the course of hours and no avail. :(**

Mod note: LaTeX fixed

[tex]\left(x+\frac{b}{a} \right)^2=-\frac{c}{a}+\frac{b^2}{2a^2}[/tex]

what I don't understand here is why the a has a degree of 2.

[tex]x+\frac{b}{a} =\pm \sqrt{-\frac{c}{a}+\frac{b^2}{2a^2}}[/tex]

ugh...

[tex]x=\frac{b}{a} \pm \sqrt{-\frac{c}{a}+\frac{b^2}{2a^2}}[/tex]

[tex]x=\frac{b}{a} \pm \sqrt{ \frac{b^2}{2a^2}+\frac{c}{a}}[/tex]

These two, are where I get stuck, I don't understand how to form the [tex]b^2-4ac[/tex] in the radicand, I don't know how to form the [tex]2a[/tex] in the denominator, I don't know how the [tex]-b /pm [/tex] ever gets there. I just don't know how to derive the quadratic formula, though, I have tried many times in the past day and a half, I eventually forfeited, and realized that I may need some help figuring this out, so I came to the online community I thought would be best at helping me out with this, and that is physics forums, if you help me understand this, thanks in advance.

If you help me understand my second question, I'll more than appreciate you, I'll remember your name as well! I have been on these forums for a bit of time now, I rarely, rarely RARELY post. but that's going to change starting today, I'm going to hone my skills by asking for help, and I will contribute wherever I can, my goal is to help for every-time I get help, and to help 1000 times, lol, a hefty goal. But I got some bad karma that needs fixing, let's just leave it at that,

so here is the second question, it's about the definition of derivation, and understanding the concept of it, what it's used for, and how it works; and why it was developed. The history of derivation and the common pattern that is understood for every rule of derivation, for now, I only know the power rule of derivation, and I know that Gottfried Wilhelm von Leibniz notation d/dx or dy/dx doesn't indicate division.

at this point I'm trying my best to solve for x, I think one of my problems is that, my expierence and know-how when it comes to rational equations, radicals, rational exponents, and even factoring and complex numbers is kind of bad. With the exception of factoring, my factoring is par enough to do this, as far as working with rational/radical expressions as in this problem, I go off my gut/intuition, what I know, my past experience and what I think logically fits best, and I don't think it succeeds to say the least, lol.

My gut, and my intuition just knows this is wrong, which is why I came here, so I can be told why, and so I can make sense of this problem. It would do wonders for me in my mathematical journey, my fellow thinkers.

This is where I run into problems, now I know what it factors to, what I do not understand, is why it factors to it, it just feels like there is an x that is just sort of being tossed out the window, and as far as simplifying the left side, I struggling with that as well, I go off what I figure is logically correct but it just turns out to be wrong, can someone point out what I'm doing wrong? :( this is a good opportunity for advanced noobs to finally contribute on the physics forums, because we all know you asked more questions then helped solved, and that being a senior here requires that your stats equate to the opposite! ;D

**4. Further, questions. WINKWINK. ;D**

In both a general and mathematical sense, what is the absolute best definition of derivation? that is to say, what is the best way for a noob like me to understand what derivation is? I live by the philosophy that doing is understanding, I know how to derive a power equation, x^2 = 2x. I understand that, but I only understand it by fallowing rules that I memorized about it, I don't understand exactly why it's important, or why people decided to start deriving and how they used it in their life, I know that to derive to a conclusion, is to be presented with a question, usually a problem, and to come up with a solution. For instance, I'm lost, therefor I derived by logical consequence, a way to find my way home based on what I know and the means to use and work with what I know. but I still do not understand derivation, let alone "integration", aka the INTEGRAL. I understand limits, I've evaluated quite a few. Which is surprising because I'm just now retaking pre-calculus. And this is the very first thing they are teaching the class, is reviewing past concepts, such as working with radicals, rational expressions, complex numbers, and of course, quadratic equations.

But I'm a thirsty person, I thrive on satisfying my curiiosity I always have, can someone teach this humbled musician what it means to derive? and what an integral is? Teach me, CONCEPTUALLY. I beg you!

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