Complete the square, doesn't make sense, QUESTION(s) explain/show?

Question

x²+2ax=b
work:
(x+a)²-2a=b

This would make sense, we would have to complete the square by mutiplying "2" and then squaring it.
(x+a)²-4a²=b
(x+a)²=4a²+b
x+a=√4a²+b
x+a=2a√b
then..
x=a√b

How would it make sense skiping by mutiplying "2", and replacing it by ² to get -a² instead of just 4a²?
(x+a)²-a²=b

see?

Answer 1

Try working the problem forwards instead:

(x+a)² + k = b

We want to know what k should be. Expand:

x² + 2ax + a² + k = b

In order for there to be no constant term on the left, you want to have k = -a² so that they cancel and leave:

x² + 2ax + a² - a² = b

x² + 2ax = b

Make sense?

Answer 2

When you're completing the square you want to create a square trinomial, squaring an equation causes some problems in the ends with extra solutions. So let's do it together:

x²+2ax=b

Here you need to add a certain value that makes the left side a square trinomial. To do this simply take half of the coefficient of x and square it, and then add it on:

x²+2ax + (2a/2)² = b + (2a/2)² >>> Got to keep it equal right?
x²+2ax + a² = b + a²

Now you can factor the square trinomial:

(x+a)² = b+a²

There you go, you just completed the square.

Answer 3

x² + (2a) x = b
x² + (2a) x + a² = b + a²
(x + a)² = a² + b
x + a = ±√(a² + b)
x = - a ± √(a² + b)