To complete the square for ax²+bx you first divide by a: a(x²+bx/a).
Then we halve the x term coefficient: b/2a and square it: b²/4a² and we get a(x²+bc/a+b²/4a²). To maintain a balanced expression we must subtract this: a(x²+bc/a+b²/4a²)-a(b²/4a²). This is now the same as a(x+b/2a)²-b²/4a.
To summarise: ax²+bx = a(x+b/2a)²-b²/4a.
