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You should recall a method called completing the square from your studies of solving quadratic equations.

To refresh it in your minds we review the algebraic method of completing the square.

How can we think about completing the square algebraically?

Given an expression of the form ax2 + bx, you want to determine the appropriate number to add to the expression so that it will become an expression of the form a(x + p)2, where p is a real number that depends on a and b in a way we will figure out.

It is much easier to follow the calculations when a = 1. So let's see how to create (x + p)2 out of x2 + bx.

Working backwords is often a good strategy, especially if you do it off the record on scrap paper. People think you are a genius when you pull the right number out of the air. Start with the target (x + p)2 and expand it out to get x2 + 2px + p2. Notice the coefficient of x is 2p. In our starting expression, the coefficient of x is b. So the only way things could work out is if we used p = b/2. We need to add in the p2 = b2/4 to "complete the square".

Remember, anything you add into the expression has to be subtracted as well so you don't change the total value.

Here is the calculation you show. Everyone will be impressed with your stroke of genius at the first step. x2 + bx = x2 + bx + b2/4 - b2/4 = (x2 + bx + b2/4) - b2/4 = (x + b/2)2 - b2/4