First off, write your equation in the form:
with a, b and c being your constant coefficients.
First, we will factorise out a such that:
Now divide b by 2. Then write
If you multiply this out you should get a(x^2 +(b/a)x+d)
Then times everything by a again to give ax^2+bx+ad
The final stage is to add a constant on the end which will get you from ad to c in the final expansion.
To do this we want ad+m=c where m is the constant we are trying to find. Therefore m=c-ad.
Our final answer will be:
ax^2 +bx+c=a(x+(b/2a))^2 +m .
This is much easier to see with an example:
Complete the square of 4x^2+4x=-6
We need to rearrange this to 4x^2+4x+6=0.
First we factorise out the 4:
Now we follow through the rest of the steps:
So the final answer is