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The equation provided is \( x^2 - 6x + y^2 - 16 = 0 \), and you are tasked with completing the square to write the equation in standard form, identifying the center and radius of the circle, and graphing it.

### Step-by-Step Solution:

1. **Rewrite the equation:**  
   \( x^2 - 6x + y^2 = 16 \).  
   We moved \( -16 \) to the right-hand side.

2. **Complete the square for the \( x \)-terms:**  
   The expression involving \( x \) is \( x^2 - 6x \). To complete the square:
   \[
   \text{Take half of } -6 \text{ (which is } -3), \text{ and square it } (-3)^2 = 9.
   \]
   Add \( 9 \) to both sides of the equation to maintain equality:
   \[
   x^2 - 6x + 9 + y^2 = 16 + 9,
   \]
   simplifying to:
   \[
   (x - 3)^2 + y^2 = 25.
   \]

3. **Final equation in standard form:**  
   The equation is now:
   \[
   (x - 3)^2 + y^2 = 25.
   \]
   This is the standard form of a circle equation.

4. **Identify the center and radius of the circle:**  
   - The center is \( (3, 0) \).
   - The radius is \( \sqrt{25} = 5 \).

### Conclusion:
- The equation in standard form is \( (x - 3)^2 + y^2 = 25 \).
- The center of the circle is \( (3, 0) \), and the radius is \( 5 \).

Would you like further details or any clarification?

### Related Questions:
1. What is the geometric interpretation of completing the square?
2. How do you graph a circle given its standard equation?
3. What other conic sections can be derived from quadratic equations?
4. How do you determine the center of a circle from a general quadratic equation?
5. Can completing the square be used for solving other types of equations, like quadratic inequalities?

**Tip:** Completing the square is a versatile algebraic technique used in solving quadratics, integrating circles, and deriving formulas like the quadratic formula.

Complete the square and write the given equation in standard form. Then give the center and radius of the circle and graph the equation: x^2 - 6x + y^2 - 16 = 0.

Learn how to complete the square to write the equation x^2 - 6x + y^2 - 16 = 0 in standard form, and find the center and radius of the circle. This problem covers essential concepts in algebra and geometry, including completing the square and the standard form of a circle. Suitable for Grades 9-10.

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