Question:The equation x^2 - 6x + y^2 + 8y = 11 represents a circle in the xy-plane. What is the...

1. INFER the solution strategy

• Given: \(\mathrm{x^2 - 6x + y^2 + 8y = 11}\)
• To find the square of the radius, we need this in standard form: \(\mathrm{(x - h)^2 + (y - k)^2 = r^2}\)
• Strategy: Complete the square for both x and y terms

2. SIMPLIFY by completing the square for x terms

• Start with: \(\mathrm{x^2 - 6x}\)
• Take half the coefficient of x: \(\mathrm{-6 \div 2 = -3}\)
• Square it: \(\mathrm{(-3)^2 = 9}\)
• Rewrite: \(\mathrm{x^2 - 6x = (x - 3)^2 - 9}\)

3. SIMPLIFY by completing the square for y terms

• Start with: \(\mathrm{y^2 + 8y}\)
• Take half the coefficient of y: \(\mathrm{8 \div 2 = 4}\)
• Square it: \(\mathrm{(4)^2 = 16}\)
• Rewrite: \(\mathrm{y^2 + 8y = (y + 4)^2 - 16}\)

4. SIMPLIFY by substituting back and rearranging

• Original equation: \(\mathrm{x^2 - 6x + y^2 + 8y = 11}\)
• Substitute: \(\mathrm{(x - 3)^2 - 9 + (y + 4)^2 - 16 = 11}\)
• Combine constants: \(\mathrm{(x - 3)^2 + (y + 4)^2 - 25 = 11}\)
• Add 25 to both sides: \(\mathrm{(x - 3)^2 + (y + 4)^2 = 36}\)

5. INFER the final answer

• Standard form: \(\mathrm{(x - 3)^2 + (y + 4)^2 = 36}\)
• Comparing to \(\mathrm{(x - h)^2 + (y - k)^2 = r^2}\): \(\mathrm{r^2 = 36}\)

Answer: D) 36

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students don't recognize that they need to complete the square to get standard form. Instead, they might try to factor the original equation or attempt other approaches that don't work for this form.

This leads to confusion and guessing among the answer choices.

Second Most Common Error:

Poor SIMPLIFY execution: Students make sign errors when completing the square, particularly with the y terms. For example, writing \(\mathrm{(y - 4)^2}\) instead of \(\mathrm{(y + 4)^2}\), or miscalculating the constant terms when combining -9, -16, and 11.

This may lead them to select Choice C (25) if they incorrectly get \(\mathrm{r^2 = 25}\) due to arithmetic errors.

The Bottom Line:

This problem requires both strategic thinking (recognizing the need for completing the square) and careful algebraic execution. Many students know the standard form but struggle with the systematic process of completing the square for both variables simultaneously.