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Derivation of Quadratic Formula

A Quadratic Equation looks like this:

Quadratic Equation

And it can be solved using the Quadratic Formula:

Quadratic Formula

That formula looks like magic, but you can follow the steps to see how it comes about.

1. Complete the Square

It is hard to handle an equation where "x" appears twice, but there is a way to rearrange it so that "x" only appears once. It is called "Completing the Square" (please read about it first!).

So, I am going to take advantage of what happens when you expand (x+d)2

(x+d)2 = (x+d)(x+d) = x(x+d) + d(x+d) = x2 + 2dx + d2

So, if we can get the equation into the form:
 
x2 + 2dx + d2
 
Then we can immediately rewrite it as:
 
(x+d)2
 
Which is going to make the rest of the task easier

So, let's go:
Start with
Divide the equation by a
Put c/a on other side
Add (b/2a)2 to both sides

Aha! we have the x2 + 2dx + d2 format that we wanted!
(if we treat "b/2a" as "d", that is)

"Complete the Square"

2. Now Solve For "x"

Now we just need to rearrange the equation to leave "x" on the left

Start with
Square root
Move b/2a to right

That is actually solved! But let's simplify it a bit:

Multiply right by 2a/2a
Simplify:

Which is the Quadratic formula we all know and love:

  Quadratic Formula