We will complete the square and find out what the vertex means to the problem. Start by setting the quadratic equal to 0 then move over the 11 by subtraction. That gives us [tex] 3x^2-6x=-11 [/tex]. The rule for completing the square is that the leading coefficient has to be a 1. Ours is a 3, so we have to factor it out. [tex] 3(x^2-2x)=-11 [/tex]. Now we will take half the linear term, square it, and add it to both sides. Our linear term is 2. Half of 2 is 1, and 1 squared is 1. So we add 1 inside the parenthesis on the left. But don't forget that 3 hanging around outside refusing to be ignored. It is a multiplier. What we have actually added in is 3 times 1 which is 3. So this is what that looks like: [tex] 3(x^2-2x+1)=-11+3 [/tex]. On the right we simplify to -8 but on the right we have a perfect square binomial, which is the whole reason for what we have done. That perfect square binomial is [tex] 3(x-1)^2=-8 [/tex]. If we move the -8 over by addition we have [tex] 3(x-2)^2+8=y [/tex]. That tells us that the vertex is (1, 8). If x is the number of items produced and y is the cost, our vertex tells us that the minimum cost to produce that 1 (x) item is $8 (y). Your answer is the first one listed.
