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

ax2 + bx + c has "x" in it twice, which is hard to solve.

But there is a way to rearrange it so that "x" only appears once. It is called Completing the Square (please read that first!).

Our aim is to get something like x2 + 2dx + d2, which can be simplified to (x+d)2

So, let's go:

Start with
Divide the equation by a
Put c/a on other side
Add (b/2a)2 to both sides

The left hand side is now in the x2 + 2dx + d2 format, where "d" is "b/2a"
So we can re-write it this way:

"Complete the Square"

Now x only appears once and we are making progress.

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