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y=x2-18x+52
First off, this does not factor cleanly, but even if it did it would not help us. We must complete the square to yield a perfect-square trinomial (a trinomial that can be written in the form (x-a)2 where a is a real number.
To do this, halve the coefficient of the single-x term (for this case, the term 18x) and square it.
18/2=9
92=81
The result of this process can be added to the two terms involving x to get a perfect square trinomial. For instance, the trinomial that would result would be:
x2-18x+81=(x-9)2
However, there is already 52 of that 81 that is needed present, so in reality, you only need to add 29 (81-52=29) to the right side of the equation to get a perfect-square trinomial. Whatever is done to one side of the equation must be done to the other side, so since you added 29 to the right side, you must add 29 to the left side, giving you:
y+29=x2-18x+52+29
y+29=x2-18x+81
y+29=(x-9)2
y=(x-9)2-29
This is in vertex form, you can see that the vertex is at (9,-29)
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