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: The equation \(\mathrm{x^2 + x + y^2 + y = \frac{199}{2}}\) represents a circle
• Find: The length of the circle's radius

2. INFER the approach needed

• The equation is in general form, but we need standard form to identify the radius
• Standard circle form is: \(\mathrm{(x - h)^2 + (y - k)^2 = r^2}\)
• We need to complete the square for both x and y terms to convert the equation

3. SIMPLIFY by completing the square for x terms

• Start with: \(\mathrm{x^2 + x}\)
• To complete the square: take half the coefficient of x, then square it
• Coefficient of x is 1, so half is \(\mathrm{\frac{1}{2}}\), and \(\mathrm{(\frac{1}{2})^2 = \frac{1}{4}}\)
• Therefore: \(\mathrm{x^2 + x = (x + \frac{1}{2})^2 - \frac{1}{4}}\)

4. SIMPLIFY by completing the square for y terms

• Start with: \(\mathrm{y^2 + y}\)
• Same process: coefficient of y is 1, so \(\mathrm{(\frac{1}{2})^2 = \frac{1}{4}}\)
• Therefore: \(\mathrm{y^2 + y = (y + \frac{1}{2})^2 - \frac{1}{4}}\)

5. SIMPLIFY by substituting and rearranging

• Original equation becomes:
  \(\mathrm{(x + \frac{1}{2})^2 - \frac{1}{4} + (y + \frac{1}{2})^2 - \frac{1}{4} = \frac{199}{2}}\)
• Combine the constant terms on the left:
  \(\mathrm{(x + \frac{1}{2})^2 + (y + \frac{1}{2})^2 - \frac{1}{2} = \frac{199}{2}}\)
• Add \(\mathrm{\frac{1}{2}}\) 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

• The equation \(\mathrm{(x + \frac{1}{2})^2 + (y + \frac{1}{2})^2 = 100}\) is in standard form
• Comparing with \(\mathrm{(x - h)^2 + (y - k)^2 = r^2}\), we see \(\mathrm{r^2 = 100}\)
• Therefore: \(\mathrm{r = \sqrt{100} = 10}\)

Answer: 10

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY execution: Students make arithmetic errors when combining fractions or completing the square incorrectly.

For example, they might add \(\mathrm{\frac{199}{2} + \frac{1}{4} + \frac{1}{4}}\) incorrectly, getting \(\mathrm{\frac{200}{4} = 50}\) instead of \(\mathrm{\frac{200}{2} = 100}\). This would lead to \(\mathrm{r^2 = 50}\), giving \(\mathrm{r = \sqrt{50} \approx 7.07}\). This leads to confusion since no such answer choice exists, causing them to guess.

Second Most Common Error:

Missing conceptual knowledge about completing the square: Students may not remember that to complete the square for \(\mathrm{x^2 + bx}\), they need to add \(\mathrm{(\frac{b}{2})^2}\).

They might try other approaches like factoring or incorrectly assume the equation is already in standard form. This causes them to get stuck and abandon systematic solution, leading to guessing.

The Bottom Line:

This problem tests whether students can systematically apply the completing the square technique and carefully handle fraction arithmetic. The key insight is recognizing that the general form must be converted to standard form before the radius can be identified.