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The given equation of the circle is:

\[
x^2 + 6x + y^2 - 8y = 0
\]

To find the center and the radius of the circle, we will rewrite this equation in standard form by completing the square for both \(x\) and \(y\) terms.

### Step 1: Group the \(x\) and \(y\) terms together
\[
(x^2 + 6x) + (y^2 - 8y) = 0
\]

### Step 2: Complete the square

- For the \(x\)-terms:
  \[
  x^2 + 6x
  \]
  Take half of the coefficient of \(x\) (which is 6), square it, and add it inside the parentheses. Half of 6 is 3, and \(3^2 = 9\). So, add and subtract 9:
  \[
  (x^2 + 6x + 9 - 9) = (x + 3)^2 - 9
  \]

- For the \(y\)-terms:
  \[
  y^2 - 8y
  \]
  Take half of the coefficient of \(y\) (which is -8), square it, and add it inside the parentheses. Half of -8 is -4, and \((-4)^2 = 16\). So, add and subtract 16:
  \[
  (y^2 - 8y + 16 - 16) = (y - 4)^2 - 16
  \]

### Step 3: Rewrite the equation
Now, substitute the completed squares back into the equation:
\[
(x + 3)^2 - 9 + (y - 4)^2 - 16 = 0
\]

### Step 4: Simplify
Combine constants on the right-hand side:
\[
(x + 3)^2 + (y - 4)^2 = 25
\]

### Step 5: Interpret the standard form
The equation is now in the standard form of a circle:
\[
(x - h)^2 + (y - k)^2 = r^2
\]
Where \((h, k)\) is the center and \(r\) is the radius.

From the equation \((x + 3)^2 + (y - 4)^2 = 25\), we can see that:
- The center of the circle is \((-3, 4)\).
- The radius is \(\sqrt{25} = 5\).

### Final Answer:
- **Center:** \((-3, 4)\)
- **Radius:** \(5\)

Would you like any further details or explanations?

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Here are five related questions to expand your understanding:
1. How do you complete the square for a general quadratic equation?
2. What is the geometric interpretation of the equation of a circle?
3. How do you convert an ellipse equation into standard form?
4. What is the general formula for the equation of a circle in 3D?
5. How does the radius of a circle change if you alter its equation?

**Tip:** When completing the square, always remember to balance both sides of the equation by adding or subtracting constants accordingly.

Find the center and the radius of the following circle: x^2 + 6x + y^2 - 8y = 0

Discover how to determine the center and radius of a circle by rewriting the equation x^2 + 6x + y^2 - 8y = 0 into its standard form. This guide explains the process of completing the square for both x and y variables, identifying the circle's center at (-3, 4) and its radius as 5. Ideal for high school level geometry.

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