To convert the given equation from standard form to graphing form, we will use the method of completing the square.
The given equation is:
[tex]\[ x^2 - 6x + y^2 - 4y + 12 = 0 \][/tex]
### Step-by-Step Solution
1. Group the [tex]\(x\)[/tex] terms and the [tex]\(y\)[/tex] terms together:
[tex]\[ (x^2 - 6x) + (y^2 - 4y) + 12 = 0 \][/tex]
2. Complete the square for the [tex]\(x\)[/tex] terms:
[tex]\[ x^2 - 6x = (x - 3)^2 - 9 \][/tex]
3. Complete the square for the [tex]\(y\)[/tex] terms:
[tex]\[ y^2 - 4y = (y - 2)^2 - 4 \][/tex]
4. Substitute these squares back into the equation:
[tex]\[ (x - 3)^2 - 9 + (y - 2)^2 - 4 + 12 = 0 \][/tex]
5. Simplify the constant terms:
[tex]\[ (x - 3)^2 + (y - 2)^2 - 9 - 4 + 12 = 0 \][/tex]
[tex]\[ (x - 3)^2 + (y - 2)^2 - 1 = 0 \][/tex]
6. Move the constant term to the right side of the equation:
[tex]\[ (x - 3)^2 + (y - 2)^2 = 1 \][/tex]
The converted equation is now in graphing form:
[tex]\[ (x - 3)^2 + (y - 2)^2 = 1 \][/tex]
This matches the option:
c [tex]\((x - 3)^2 + (y - 2)^2 = 1\)[/tex]
Thus, the correct answer is:
[tex]\[ \boxed{\text{c}} \][/tex]
