Behold:

x = (-b ^{+}/_{-} √b^{2} - 4ac)/(2a)

Now Mathimoto's Complaint aims to cover a wide range of fans of math and today I'd like to reach out to the younger crowd. That precious younger crowd undoubtedly have seen the beauty of the quadric equation. But they and probably some older folks as well have never given an effort at deriving it. Well, I thought I'd give it a shot and show you the awesomeness of figuring out these formulas. Because Math rocks, it really does.

First start out with the generalized form of a basic quadratic equation (ie any equation with 1 variable (let's say x) and some instance of that variable raised to power 2 and possibly some instance of that variable raised to power 1. Okay, so it was harder to describe things rather than write it out, so let's do that)

Let a, b, and c be constants and x be a variable.

A basic general quadratic equation is:

ax^{2} + bx + c = 0

Now it helps then to know one particular quadratic equation, that is what happens when you have

(x + b)^{2} = 0

x^{2} + 2bx + b^{2} = 0

(Don't believe me, just use the distributive property of multiplication, ie,

(x+b)^{2} = (x + b)(x + b) =

x (x + b) + b (x + b) = x^{2} + bx + bx + b^{2} =

x^{2} + 2bx + b^{2}

Okay, now if you got a quadratic equation of the form

x^{2} + 2bx + b^{2} = 0

And you know

(x + b)^{2} = x^{2} + 2bx + b^{2}

(x + b)^{2} = 0

x = -b

Back to the general basic quadratic equation:

ax^{2} + bx + c = 0

At this point we don't know how to find the solution value of x here, but since we know the solution for

(x + b)^{2}

If

ax^{2} + bx + c = 0

ax^{2} + bx = -c

x^{2} + ^{bx}/_{a} = ^{-c}/_{a}

*(x + b) ^{2} = x^{2} + 2bx + b^{2}*

*(x + d) ^{2} = x^{2} + 2dx + d^{2}*

x^{2} + ^{bx}/_{a} = ^{-c}/_{a}

*d = ^{b}/_{2a}*

x^{2} + 2dx = ^{-c}/_{a}

^{2}and we can just add that to both sides, so:

x^{2} + 2dx + d^{2}= ^{-c}/_{a} + d^{2}

*(x + d) ^{2} = x^{2} + 2dx + d^{2}*

(x + d)^{2}= ^{-c}/_{a} + d^{2}

x + d = ^{+}/_{-} √d^{2} - ^{c}/_{a}

x = -d ^{+}/_{-} √d^{2} - ^{c}/_{a}

*d = ^{b}/_{2a}*

x = ^{-b}/_{2a} ^{+}/_{-} √(^{b}/_{2a})^{2} - ^{c}/_{a}

x = (-b ^{+}/_{-} √b^{2} - 4ac)/(2a)

And there we go, we've got the quadratic formula! Yaaaah!!!! Behold it and be amazed!!!

x = (-b ^{+}/_{-} √b^{2} - 4ac)/(2a)

MATH RULES!!!! WOOOOO!!!!

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