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To solve the problem, we are given the equation of a circle in the form:

\[
x^2 + x + y^2 + y = \frac{199}{2}
\]

We need to find the radius of the circle. 

### Step 1: Complete the Square

The general equation of a circle is:

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

where \((h, k)\) is the center and \(r\) is the radius. To rewrite the given equation in this form, we need to complete the square for both the \(x\) and \(y\) terms.

1. **For the \(x\) terms:**
   \[
   x^2 + x
   \]
   Add and subtract \(\left(\frac{1}{2}\right)^2 = \frac{1}{4}\) inside the equation to complete the square:
   \[
   x^2 + x = \left(x + \frac{1}{2}\right)^2 - \frac{1}{4}
   \]

2. **For the \(y\) terms:**
   \[
   y^2 + y
   \]
   Similarly, add and subtract \(\left(\frac{1}{2}\right)^2 = \frac{1}{4}\) to complete the square:
   \[
   y^2 + y = \left(y + \frac{1}{2}\right)^2 - \frac{1}{4}
   \]

### Step 2: Substitute and Simplify

Substituting back into the equation:

\[
\left(x + \frac{1}{2}\right)^2 - \frac{1}{4} + \left(y + \frac{1}{2}\right)^2 - \frac{1}{4} = \frac{199}{2}
\]

Simplify by combining constants on the right-hand side:

\[
\left(x + \frac{1}{2}\right)^2 + \left(y + \frac{1}{2}\right)^2 = \frac{199}{2} + \frac{1}{4} + \frac{1}{4}
\]

\[
\left(x + \frac{1}{2}\right)^2 + \left(y + \frac{1}{2}\right)^2 = \frac{199}{2} + \frac{2}{4} = \frac{199}{2} + \frac{1}{2} = \frac{200}{2} = 100
\]

### Step 3: Identify the Radius

The equation is now in the standard form of a circle's equation:

\[
\left(x + \frac{1}{2}\right)^2 + \left(y + \frac{1}{2}\right)^2 = 100
\]

Here, \(100\) is the square of the radius (\(r^2\)), so the radius \(r\) is:

\[
r = \sqrt{100} = 10
\]

### Final Answer

The radius of the circle is \(10\).

Would you like a detailed explanation of any part, or do you have any further questions?

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Here are some related questions:

1. How do you complete the square for quadratic expressions?
2. Can you explain the general form of a circle's equation?
3. What does the center of the circle represent in the equation \((x-h)^2 + (y-k)^2 = r^2\)?
4. How would the radius change if the equation were different?
5. How do transformations affect the radius and center of a circle in its equation?
6. Can the circle equation ever represent a different geometric shape?
7. How would you graph a circle given its equation?
8. What are some real-life applications of circle equations?

**Tip:** When completing the square, remember to balance the equation by adding the same value to both sides.

Learn how to solve the equation of a circle x^2 + x + y^2 + y = 199/2 using the completing the square method. This problem involves completing the square for quadratic expressions and finding the radius of a circle. Suitable for advanced high school students.

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