INSTRUCTOR: Find the equation of the level surface of the function g of x comma y comma z equals x squared plus y squared plus z squared minus 2x plus 4y minus 6z corresponding to c equals 2 and describe the surface if possible. So g of xyz, going to group my x terms together. X squared minus 2x, leave a little bit of space, plus y squared plus y, leave a little space plus z squared minus. Z

Now I want to complete the square for each of these in each variable. So x squared minus 2x plus 1 would be a perfect square. I'm going to add 1 and then also subtract 1 right there. Y squared plus 4y plus 4 v a perfect square. So I'm going to add 4 and subtract 4. And the z squared minus z plus 9 would be a perfect square. So I'm also going to subtract 9.

So g of xyz is x squared minus 2x plus 1 minus 1, plus y squared plus 4y plus 4 minus 4, plus z squared minus z plus 9 minus 9. Each of my perfect squares will factor. The first is going to be parentheses x minus 1, close parentheses, squared plus y plus 2 in parentheses, squared plus, parentheses z minus 3 close parentheses, squared.

And then I'm going to combine all those terms-- the extra minus 1 minus 4 and minus 9 that I added. That's minus 14, minus 14. And that equals our C value, which is equal to 2 in this case. So if we change things around what we have is the equation parentheses x minus 1 squared plus, parentheses y plus 2 squared plus, parentheses z minus 3 close parentheses squared equals 16. And this is a sphere of radius 4 centered at 0.1 comma negative 2 comma 3. And that is the level surface.