Question

completing the sqare method ​

Answer 1

Say we have a simple expression like x2 + bx. Having xtwice in the same expression can make life hard. What can we do?
Well, with a little inspiration from Geometry we can convert it, like this:

As you can see x2 + bx can be rearranged nearly into a square ...
... and we can complete the square with (b/2)2
In Algebra it looks like this:
x2 + bx+ (b/2)2=(x+b/2)2 "Complete the Square"  
So, by adding (b/2)2 we can complete the square.
And (x+b/2)2 has x only once, which is easier to use.

Keeping the Balance

Now ... we can't just add (b/2)2 without also subtracting it too! Otherwise the whole value changes.
So let's see how to do it properly with an example:
Start with:   ("b" is 6 in this case)  Complete the Square:
 

Also subtract the new term
Simplify it and we are done.
 
The result:
x2 + 6x + 7   =   (x+3)2 − 2
And now x only appears once, and our job is done!

A Shortcut Approach

Here is a quick way to get an answer. You may like this method.
First think about the result we want: (x+d)2 + e
After expanding (x+d)2 we get: x2 + 2dx + d2 + e
Now see if we can turn our example into that form to discover d and e