Given the expression:
-x^2+18x-99
to solve by completing squares we proceed as follows:
-x^2+18x-99=0
this can be written as:
-x^2+18x=99
x^2-18x=-99.......i
but
c=(-b/2)²
Hence:
c=(-(-18)/2)²=81
adding 81 in both sides of i we get:
x^2-18x+81=-99+81
factorizing the quadratic we obtain:
(x-9)(x-9)=-18
thus
(x-9)²+18=0
the above takes the vertex form of :
y=(x-k)²+h
where (k,x) is the vertex:
the vertex of our expression is:
(9,18)
hence the maximum point is at (9,18)
NOTE: The vertex gives the maximum point because, from the expression we see that the coefficient of the term that has the highest degree is a negative, and since our polynomial is a quadratic expression then our graph will face down, and this will make the vertex the maximum point.