Note that the general vertex form is
#color(white)("XXX")y=color(green)(m)(x-color(red)(a))^2+color(blue)(b)#
for a parabola with vertex at #(color(red)a,color(blue)b)#
Given
#color(white)("XXX")y=-x^2+12x-11#
Extract the #color(green)(m)# factor from the first two terms:
#color(white)("XXX")y=color(green)(""(-1))(x^2-12x) -11#
If #(x^2-12x)# are the first two terms of a squared binomial the third term must be #6^2#
[since #(x-a)^2=(x^2-2ax+a^2)]#
and anything we add to complete the square must also be subtracted.
Completing the square
#color(white)("XXX")y=color(green)(""(-1))(x^2-12xcolor(magenta)(+6^2))-11 color(magenta)(-color(green)(""(-1)) * (6^2))#
Simplifying:
#color(white)("XXX")y=color(green)(""(-1))(x-color(red)(6))^2+color(blue)(25)#
Here is a graph of the original equation to help verify this result:
graph{-x^2+12x-11 [-3.106, 12.697, 18.77, 26.67]}
