What is the vertex form of and how did you get the answer?

so, x²-6x + 11, let's do some grouping first

( x² - 6x ) +11

( x² - 6x + b² ) +11

now, what's our mystery fellow "b"?  well, let's recall that the middle term in a perfect square trinomial is, (a - b)² = a² - 2ab + b², so the middle term is namely just 2 times the other two fellows, without the exponent.

now, here is 6x, so then  

that means, that our fellow is 3 then, so we'll add 3², however, let's keep in mind that, all we're doing is, borrowing from our very good friend Mr Zero, 0.

so if we add 3², we have to also subtract 3², therefore,