MathsIsFun.com

A Quadratic Equation looks like this:

And it can be solved using the 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: