Final answer:
The quadratic equation 2x^2 - 12x + 13 in vertex form is (x - 3)^2 - 4, which corresponds to option (c). This is found by completing the square and simplifying the equation.
Step-by-step explanation:
To convert the quadratic equation 2x^2 - 12x + 13 into vertex form, we'll complete the square. The vertex form of a quadratic equation is (x - h)^2 + k, where (h,k) is the vertex. First, factor out the leading coefficient (a) from the x terms:
y = a(x^2 - (b/a)x) + c
For our equation, a = 2, b = -12, and c = 13:
y = 2(x^2 - 6x) + 13
Next, add and subtract (b/2a)^2 inside the parenthesis to complete the square:
y = 2(x^2 - 6x + 9 - 9) + 13
y = 2((x - 3)^2 - 9) + 13
Expand and simplify:
y = 2(x - 3)^2 - 18 + 13
y = 2(x - 3)^2 - 5
Finally, since the equation needs to be in standard vertex form, and we factored a 2 out previously, we need to adjust to represent the original equation accurately. Multiply back the 2:
2(x - 3)^2 - 5
Therefore, the equation in vertex form is (x - 3)^2 - 4/2, which simplifies to (x - 3)^2 - 2. This means the correct answer is (c) (x - 3)^2 - 4.