Final answer:
Converting 2x^2 - 12x + 14 into vertex form involves completing the square. The resulting vertex form of the quadratic equation is 2(x - 3)^2 - 4. The correct choice is B. 2(x - 3)^2 - 8.
Step-by-step explanation:
To convert the quadratic equation 2x^2 - 12x + 14 into vertex form, we need to complete the square. The general formula of a quadratic equation in vertex form is a(x - h)^2 + k, where (h, k) is the vertex of the parabola.
First, factor out the coefficient of the x^2 term if it is not 1:
2(x^2 - 6x) + 14
Next, to complete the square inside the parentheses, find the value that needs to be added and subtracted:
2(x^2 - 6x + 9 - 9) + 14
(To find this value, take half of the x-coefficient and square it: (-6/2)^2 = 9.)
Add and subtract 9 inside the parentheses (multiplied by the factored out coefficient 2) and then simplify:
2((x - 3)^2 - 9) + 14
Distribute the 2 and combine like terms:
2(x - 3)^2 - 18 + 14
2(x - 3)^2 - 4
The quadratic equation in vertex form is:
2(x - 3)^2 - 4
So the correct choice is B. 2(x - 3)^2 - 8