Question:The equation x^2 + 3x + y^2 - 5y = 6 represents a circle in the xy-plane. The center of...

1. TRANSLATE the problem information

• Given: \(\mathrm{x^2 + 3x + y^2 - 5y = 6}\) represents a circle
• Find: The k-coordinate of the center (h, k)

2. INFER the solution strategy

• To find the center, I need to convert this to standard circle form: \(\mathrm{(x - h)^2 + (y - k)^2 = r^2}\)
• This requires completing the square for both x and y terms
• Once in standard form, I can directly read off the center coordinates

3. SIMPLIFY by grouping like terms

• Separate x and y terms: \(\mathrm{(x^2 + 3x) + (y^2 - 5y) = 6}\)
• This sets up for completing the square on each variable separately

4. SIMPLIFY by completing the square for x

• For \(\mathrm{x^2 + 3x}\): take the coefficient of x (which is 3)
• Divide by 2: \(\mathrm{3 \div 2 = \frac{3}{2}}\)
• Square this result: \(\mathrm{\left(\frac{3}{2}\right)^2 = \frac{9}{4}}\)
• Add and subtract this inside the parentheses conceptually, but we'll add it to both sides of the equation

5. SIMPLIFY by completing the square for y

• For \(\mathrm{y^2 - 5y}\): take the coefficient of y (which is -5)
• Divide by 2: \(\mathrm{-5 \div 2 = -\frac{5}{2}}\)
• Square this result: \(\mathrm{\left(-\frac{5}{2}\right)^2 = \frac{25}{4}}\)

6. SIMPLIFY the equation with completion terms

• Add both completion terms to the right side:

  \(\mathrm{\left(x^2 + 3x + \frac{9}{4}\right) + \left(y^2 - 5y + \frac{25}{4}\right) = 6 + \frac{9}{4} + \frac{25}{4}}\)

• Factor the left side: \(\mathrm{\left(x + \frac{3}{2}\right)^2 + \left(y - \frac{5}{2}\right)^2 = 6 + \frac{34}{4}}\)
• SIMPLIFY the right side (use calculator): \(\mathrm{6 + \frac{34}{4} = \frac{24}{4} + \frac{34}{4} = \frac{58}{4} = \frac{29}{2}}\)

7. INFER the center coordinates

• The equation is now in standard form: \(\mathrm{\left(x + \frac{3}{2}\right)^2 + \left(y - \frac{5}{2}\right)^2 = \frac{29}{2}}\)
• Comparing with \(\mathrm{(x - h)^2 + (y - k)^2 = r^2}\):
• \(\mathrm{x + \frac{3}{2} = x - \left(-\frac{3}{2}\right)}\), so \(\mathrm{h = -\frac{3}{2}}\)
• \(\mathrm{y - \frac{5}{2} = y - \left(\frac{5}{2}\right)}\), so \(\mathrm{k = \frac{5}{2} = 2.5}\)

Answer: B) 2.5

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY skill: Students make sign errors when completing the square, especially with the y-terms. They might think \(\mathrm{y^2 - 5y}\) requires adding \(\mathrm{\left(-\frac{5}{2}\right)^2}\) but then use \(\mathrm{+\frac{5}{2}}\) in the factored form instead of \(\mathrm{-\frac{5}{2}}\), or they might incorrectly compute the completion values.

This typically leads to getting \(\mathrm{k = -2.5}\) instead of \(\mathrm{+2.5}\), but since -2.5 isn't among the choices, this leads to confusion and guessing.

Second Most Common Error:

Poor INFER reasoning: Students recognize they need to complete the square but don't understand how the standard form relates to the center. They might complete the square correctly but then misread the center coordinates, thinking \(\mathrm{\left(x + \frac{3}{2}\right)^2}\) means \(\mathrm{h = +\frac{3}{2}}\) instead of \(\mathrm{h = -\frac{3}{2}}\).

Since the question only asks for k, they might still get \(\mathrm{k = \frac{5}{2} = 2.5}\) correctly despite the h-coordinate error.

The Bottom Line:

This problem tests both procedural fluency with completing the square and conceptual understanding of how standard form reveals geometric properties. The fraction arithmetic and multiple algebraic steps create opportunities for computational errors while the sign interpretation requires careful attention to the relationship between algebraic and geometric representations.