The graph of x^2 + x + y^2 + y = 199/2 in the xy-plane is a circle. What is...

1. TRANSLATE the problem information

• Given: \(\mathrm{x^2 + x + y^2 + y = \frac{199}{2}}\) represents a circle
• Find: The radius length

2. INFER the solution approach

• The given equation is in general form, but we need it in standard form \(\mathrm{(x - h)^2 + (y - k)^2 = r^2}\) to find the radius
• Strategy: Complete the square for both x and y terms to convert to standard form

3. SIMPLIFY by completing the square for x terms

• Start with \(\mathrm{x^2 + x}\)
• Take half the coefficient of x: \(\mathrm{1 ÷ 2 = \frac{1}{2}}\)
• Square this value: \(\mathrm{(\frac{1}{2})^2 = \frac{1}{4}}\)
• Add and subtract this: \(\mathrm{x^2 + x + \frac{1}{4} = (x + \frac{1}{2})^2}\)

4. SIMPLIFY by completing the square for y terms

• Start with \(\mathrm{y^2 + y}\)
• Same process: half of 1 is \(\mathrm{\frac{1}{2}}\), squared gives \(\mathrm{\frac{1}{4}}\)
• So: \(\mathrm{y^2 + y + \frac{1}{4} = (y + \frac{1}{2})^2}\)

5. SIMPLIFY the equation by adding completion constants to both sides

• Original: \(\mathrm{x^2 + x + y^2 + y = \frac{199}{2}}\)
• Add \(\mathrm{\frac{1}{4} + \frac{1}{4} = \frac{1}{2}}\) to both sides:
• \(\mathrm{(x + \frac{1}{2})^2 + (y + \frac{1}{2})^2 = \frac{199}{2} + \frac{1}{2} = \frac{200}{2} = 100}\)

6. INFER the radius from standard form

• We now have: \(\mathrm{(x + \frac{1}{2})^2 + (y + \frac{1}{2})^2 = 100}\)
• This matches \(\mathrm{(x - h)^2 + (y - k)^2 = r^2}\) where \(\mathrm{r^2 = 100}\)
• Therefore: \(\mathrm{r = \sqrt{100} = 10}\)

Answer: 10

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students don't recognize they need to complete the square to find the radius. They might try to directly identify the radius from the given form or attempt to factor the equation unsuccessfully. This leads to confusion and guessing.

Second Most Common Error:

Poor SIMPLIFY execution: Students attempt to complete the square but make arithmetic errors, particularly when adding fractions (forgetting that \(\mathrm{\frac{1}{4} + \frac{1}{4} = \frac{1}{2}}\)) or when calculating \(\mathrm{\frac{199}{2} + \frac{1}{2} = 100}\). These calculation mistakes lead to incorrect values for \(\mathrm{r^2}\).

The Bottom Line:

This problem requires recognizing that converting from general form to standard form via completing the square is the key strategy, then executing that multi-step algebraic process accurately.