The graph of x^2 - 4x + y^2 + 6y - 12 = 0 in the xy-plane is a circle....

1. INFER the solution strategy

• Given: \(\mathrm{x^2 - 4x + y^2 + 6y - 12 = 0}\) in general form
• Need: radius length
• Strategy: Convert to standard form \(\mathrm{(x - h)^2 + (y - k)^2 = r^2}\) by completing the square

2. SIMPLIFY the x terms by completing the square

• Start with \(\mathrm{x^2 - 4x}\)
• Take half the coefficient of x: \(\mathrm{-4 ÷ 2 = -2}\)
• Square it: \(\mathrm{(-2)^2 = 4}\)
• Rewrite: \(\mathrm{x^2 - 4x = (x - 2)^2 - 4}\)

3. SIMPLIFY the y terms by completing the square

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

4. SIMPLIFY by substituting back and combining constants

• Original: \(\mathrm{x^2 - 4x + y^2 + 6y - 12 = 0}\)
• Becomes: \(\mathrm{(x - 2)^2 - 4 + (y + 3)^2 - 9 - 12 = 0}\)
• Combine constants: \(\mathrm{(x - 2)^2 + (y + 3)^2 - 25 = 0}\)
• Rearrange: \(\mathrm{(x - 2)^2 + (y + 3)^2 = 25}\)

5. INFER the radius from standard form

• Standard form: \(\mathrm{(x - h)^2 + (y - k)^2 = r^2}\)
• We have: \(\mathrm{(x - 2)^2 + (y + 3)^2 = 25}\)
• Therefore: \(\mathrm{r^2 = 25}\), so \(\mathrm{r = 5}\)

Answer: 5

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY execution: Students make arithmetic errors when completing the square, particularly with signs and constant calculations.

For example, they might write \(\mathrm{y^2 + 6y = (y + 3)^2 - 6}\) instead of \(\mathrm{(y + 3)^2 - 9}\), or incorrectly combine the final constants as \(\mathrm{-4 - 9 - 12 = -19}\) instead of \(\mathrm{-25}\). This leads to wrong values for \(\mathrm{r^2}\) and ultimately an incorrect radius.

Second Most Common Error:

Missing conceptual knowledge about completing the square: Students may not remember the completing the square process or confuse the steps.

They might attempt to factor the equation directly or try to solve for x and y separately, getting stuck because the equation doesn't factor nicely in its current form. This leads to confusion and guessing.

The Bottom Line:

This problem tests your ability to systematically apply completing the square to both variables simultaneously. The key insight is recognizing that the general form must be converted to standard form to easily identify the radius.