The equation 2x^2 + 8x + 2y^2 - 12y = 10 represents a circle in the xy-plane. What is the...

1. INFER the problem strategy

• Given: \(2\mathrm{x}^2 + 8\mathrm{x} + 2\mathrm{y}^2 - 12\mathrm{y} = 10\)
• This equation has \(\mathrm{x}^2\) and \(\mathrm{y}^2\) terms with the same coefficient, suggesting it's a circle
• Strategy: Convert to standard form \((\mathrm{x} - \mathrm{h})^2 + (\mathrm{y} - \mathrm{k})^2 = \mathrm{r}^2\) by completing the square

2. SIMPLIFY by factoring out coefficients

• Factor out the 2 from both squared variable terms:
• \(2(\mathrm{x}^2 + 4\mathrm{x}) + 2(\mathrm{y}^2 - 6\mathrm{y}) = 10\)

3. SIMPLIFY by completing the square for each variable

• For \(\mathrm{x}^2 + 4\mathrm{x}\): Take half of 4, which is 2, then square it: \(2^2 = 4\)
• So \(\mathrm{x}^2 + 4\mathrm{x} = (\mathrm{x} + 2)^2 - 4\)
• For \(\mathrm{y}^2 - 6\mathrm{y}\): Take half of -6, which is -3, then square it: \((-3)^2 = 9\)
• So \(\mathrm{y}^2 - 6\mathrm{y} = (\mathrm{y} - 3)^2 - 9\)

4. SIMPLIFY by substituting and distributing

• Substitute: \(2((\mathrm{x} + 2)^2 - 4) + 2((\mathrm{y} - 3)^2 - 9) = 10\)
• Distribute: \(2(\mathrm{x} + 2)^2 - 8 + 2(\mathrm{y} - 3)^2 - 18 = 10\)
• Combine constants: \(2(\mathrm{x} + 2)^2 + 2(\mathrm{y} - 3)^2 - 26 = 10\)
• Add 26 to both sides: \(2(\mathrm{x} + 2)^2 + 2(\mathrm{y} - 3)^2 = 36\)

5. SIMPLIFY to standard form

• Divide everything by 2: \((\mathrm{x} + 2)^2 + (\mathrm{y} - 3)^2 = 18\)
• This matches standard form where \(\mathrm{r}^2 = 18\)

6. SIMPLIFY the radical

• \(\mathrm{r} = \sqrt{18} = \sqrt{9 \times 2} = 3\sqrt{2}\)

Answer: C (\(3\sqrt{2}\))

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students don't recognize this as a circle equation needing conversion to standard form. Instead, they might try to solve for x and y as separate equations or attempt to use the quadratic formula directly.

This leads to confusion and guessing among the answer choices.

Second Most Common Error:

Poor SIMPLIFY execution: Students make arithmetic errors when completing the square, particularly forgetting to subtract the constant term (like the -4 and -9) or making sign errors when distributing the 2.

For example, if they incorrectly get \((\mathrm{x} + 2)^2 + (\mathrm{y} - 3)^2 = 10\) instead of 18, they would calculate \(\mathrm{r} = \sqrt{10}\), leading them to select Choice A (\(\sqrt{10}\)).

The Bottom Line:

This problem requires both strategic insight (recognizing the circle pattern) and careful algebraic manipulation. Students must systematically complete the square for both variables while tracking multiple terms and coefficients accurately.