Question

How do you complete the square for `x^2+18x`?

Answer 1

`(x+9)^2 = x^2 + 18x +81`

In general,

`ax^2+bx+c = a(x+b/(2a))^2 + (c - b^2/(4a))`

Notice that the term added to `x` is `b/(2a)`

Answer 2

For a general form, squared binomial
`color(white)("XXXXX")` `(x+a)^2 = x^2+2ax+a^2`

So if `x^2+18x` are the first two terms of a squared binomial
`color(white)("XXXXX")`then, in the general form, `a=9` and
`color(white)("XXXXX")` `a^2 = 9^2 = 81`

Of course, if we are going to add `9^2` to the expression `x^2+18x` we are also going to have to subtract it:
`color(white)("XXXXX")` `x^2+18x`
`color(white)("XXXXX")` `= x^2+18xcolor(red)(+9^2) - color(blue)(9^2)`
`color(white)("XXXXX")` `=color(red)((x+9)^2) color(blue)(- 81)`