There are two answers to your question. The first answer is to do with how it works; the second is to do with why. The rule explaining the how is:
- We mayadd the same number to both sides of an inequality, and it remains true. In other words:
What's happened, then, is that 4 has been added to both sides of the inequality:
That was how the + 4 appeared.
But you may have asked: why should we want to add 4 to both sides of the inequality? The answer is that we need to create a term in so that we can establish the truth of .
So that's why the + 4 appeared.
The same thing happens again in the next stage, although it's not so clear this time. Let me put things the other way round, and you'll see how it works.
So, writing this the other way around, we get:
If we now add to both sides of , we get:
So that's the how. Now for the why. Why look at at all? Well, it's to do with completing the square, because, again, we need to create a term in to establish the truth of .
And to make into we need to involve , because .
Is that OK now?