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We are given the equation of a circle in expanded form:

\[
x^2 + y^2 - 20x + 14y + 28 = 0
\]

We need to convert this into the standard form of a circle equation, which is:

\[
(x - h)^2 + (y - k)^2 = r^2
\]

where \((h, k)\) is the center of the circle and \(r\) is the radius. We will complete the square for both the \(x\) and \(y\) terms.

### Step 1: Rearrange terms
Group the \(x\) terms and \(y\) terms:

\[
(x^2 - 20x) + (y^2 + 14y) = -28
\]

### Step 2: Complete the square for \(x\)
For the \(x\)-terms, we take half of \(-20\), square it, and add it inside the equation. Half of \(-20\) is \(-10\), and \((-10)^2 = 100\). So, add 100 to both sides:

\[
(x^2 - 20x + 100) + (y^2 + 14y) = -28 + 100
\]

\[
(x - 10)^2 + (y^2 + 14y) = 72
\]

### Step 3: Complete the square for \(y\)
For the \(y\)-terms, take half of \(14\), square it, and add it inside the equation. Half of \(14\) is \(7\), and \(7^2 = 49\). So, add 49 to both sides:

\[
(x - 10)^2 + (y + 7)^2 = 72 + 49
\]

\[
(x - 10)^2 + (y + 7)^2 = 121
\]

### Step 4: Identify the radius
The equation is now in standard form, \((x - 10)^2 + (y + 7)^2 = 121\), where the radius squared is 121. Therefore, the radius \(r\) is:

\[
r = \sqrt{121} = 11
\]

Thus, the radius of the circle is \( \boxed{11} \).

Would you like further details on any steps or have any additional questions?

Here are 5 follow-up questions to explore:

1. How do you complete the square for quadratic expressions?
2. What are the key features of the standard form of a circle?
3. How can you find the center of the circle from the general form equation?
4. What would the equation look like if the circle had a different radius, say \(r = 5\)?
5. How does the method change if the equation involves coefficients other than 1 for \(x^2\) and \(y^2\)?

**Tip:** Completing the square is a crucial technique not only for circles but also for solving quadratic equations!

The graph of the equation x^2 + y^2 - 20x + 14y + 28 = 0 is a circle. What is the radius of this circle?

Learn how to find the radius of a circle from the equation x^2 + y^2 - 20x + 14y + 28 = 0. This problem involves completing the square and understanding the standard form of a circle equation. Suitable for students in Grades 9-11.

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