f(x) = 2(x - 4)^2 + 32 <--- factor the perfect square trinomial and combine like terms
So now you have the function in what's called vertex form. The x-value of the vertex is opposite of what's being added to x. -4 is being added to x, so the x-value of the vertex should be +4. The y-value of the vertex is term that doesn't have x, so it's 32. Thus, the vertex is the point (4, 32). When you plug this into the original form of the function, x^2 + (8 - x)^2, it works, so you know you didn't make any mistakes. If you graph the function, you'll see that the parabola opens upward (the arrows point upward). You can tell this by looking at the function in vertex form because there is no negative in front of the term with x, 2(x - 4)^2. If there was a negative sign, -2(x - 4)^2, it would open downward. Since the parabola opens upward, you know that the vertex must be its minimum.
Hence, the coordinates of the minimum are the same as the coordinates of the vertex of the function: (4, 32).
So, how do you interpret this? What does this mean in real life? Well, it means that if you want the sum of the areas of the two squares to be the smallest they can be, you need to make a 4-inch cut. That is, cut the wire exactly in half. Notice that when you do this, the squares should be the same size.