How to complete the square of a standard form with a coefficient?

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.