We are given the equation:

$x^2 + y^2 + z^2 - 6x + 4y - 2z = 11$

Our task is to show that this equation represents a sphere and find its center and radius.

Step 1: Rearrange the equation

The general form of the equation of a sphere is:

$(x - h)^2 + (y - k)^2 + (z - l)^2 = r^2$

where $(h, k, l)$ is the center of the sphere and $r$ is the radius. To put the given equation into this form, we need to complete the square for the $x$, $y$, and $z$ terms.

Step 2: Complete the square

Let's complete the square for each variable.

1. For $x$: $x^2 - 6x = (x - 3)^2 - 9$

2. For $y$: $y^2 + 4y = (y + 2)^2 - 4$

3. For $z$: $z^2 - 2z = (z - 1)^2 - 1$

Step 3: Substitute into the equation

Substitute the completed square expressions back into the equation:

$(x - 3)^2 - 9 + (y + 2)^2 - 4 + (z - 1)^2 - 1 = 11$

Simplify:

$(x - 3)^2 + (y + 2)^2 + (z - 1)^2 - 14 = 11$

Add 14 to both sides:

$(x - 3)^2 + (y + 2)^2 + (z - 1)^2 = 25$

Step 4: Interpret the result

Now we have the equation of a sphere in standard form:

$(x - 3)^2 + (y + 2)^2 + (z - 1)^2 = 5^2$

This represents a sphere with:

• Center: $(3, -2, 1)$
• Radius: $5$

Conclusion

The equation represents a sphere with center $(3, -2, 1)$ and radius $5$.

Would you like further details on the steps? Here are some additional questions you might find useful to explore:

1. How do you complete the square in a general quadratic equation?
2. Why does completing the square help in identifying the equation of a sphere?
3. What other geometric shapes can result from similar quadratic equations?
4. How would the equation change if the sphere were shifted in another direction?
5. What is the physical interpretation of the center and radius in a real-world problem?

Tip: Always ensure that terms are properly grouped when completing the square for quadratic equations.