In the xy-plane, the graph of 4x^2 + 8x + 4y^2 - 40y = 40 is a circle. What is...

1. TRANSLATE the problem information

• Given: \(4\mathrm{x}^2 + 8\mathrm{x} + 4\mathrm{y}^2 - 40\mathrm{y} = 40\) represents a circle
• Find: Length of the diameter

2. INFER the approach

• The equation isn't in standard form \((\mathrm{x} - \mathrm{h})^2 + (\mathrm{y} - \mathrm{k})^2 = \mathrm{r}^2\)
• I need to complete the square for both x and y terms to find the radius
• Then \(\mathrm{diameter} = 2 \times \mathrm{radius}\)

3. SIMPLIFY by removing coefficients

• Divide the entire equation by 4: \(\mathrm{x}^2 + 2\mathrm{x} + \mathrm{y}^2 - 10\mathrm{y} = 10\)
• Now I can work with simpler coefficients

4. SIMPLIFY by completing the square for x terms

• For \(\mathrm{x}^2 + 2\mathrm{x}\): take half of 2 = 1, then square it = 1
• \(\mathrm{x}^2 + 2\mathrm{x} + 1 = (\mathrm{x} + 1)^2\)
• Must add 1 to the right side too

5. SIMPLIFY by completing the square for y terms

• For \(\mathrm{y}^2 - 10\mathrm{y}\): take half of -10 = -5, then square it = 25
• \(\mathrm{y}^2 - 10\mathrm{y} + 25 = (\mathrm{y} - 5)^2\)
• Must add 25 to the right side too

6. SIMPLIFY to get standard form

• \((\mathrm{x} + 1)^2 + (\mathrm{y} - 5)^2 = 10 + 1 + 25 = 36\)
• This matches \((\mathrm{x} - \mathrm{h})^2 + (\mathrm{y} - \mathrm{k})^2 = \mathrm{r}^2\) where \(\mathrm{r}^2 = 36\)

7. INFER the final answer

• Since \(\mathrm{r}^2 = 36\), then \(\mathrm{r} = 6\)
• Diameter = \(2\mathrm{r} = 2(6) = 12\)

Answer: C. 12

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY execution: Students make arithmetic errors when completing the square, especially forgetting to add the completion constants to both sides of the equation.

For example, they might complete the square for x correctly (adding 1) but forget to add 25 when completing the square for y, leading to \(\mathrm{r}^2 = 11\) instead of 36. This gives \(\mathrm{r} \approx 3.3\) and \(\mathrm{diameter} \approx 6.6\), leading them to guess Choice A (6) as the closest option.

Second Most Common Error:

Conceptual confusion about radius vs diameter: Students correctly find \(\mathrm{r} = 6\) but then forget that the problem asks for diameter, not radius.

This leads them to select Choice A (6) instead of recognizing that \(\mathrm{diameter} = 2\mathrm{r} = 12\).

The Bottom Line:

This problem tests both algebraic manipulation skills (completing the square) and careful attention to what the question actually asks for. Success requires methodical completion of the square and remembering the radius-diameter relationship.