In the xy-plane, the graph of 2x^2 - 6x + 2y^2 + 2y = 45 is a circle. What is...

1. INFER the solution strategy

• Given: \(2\mathrm{x}^2 - 6\mathrm{x} + 2\mathrm{y}^2 + 2\mathrm{y} = 45\)
• Need: radius of the circle
• Strategy: Convert to standard form \((\mathrm{x} - \mathrm{h})^2 + (\mathrm{y} - \mathrm{k})^2 = \mathrm{r}^2\) to identify r

2. SIMPLIFY by making coefficients equal to 1

• Divide the entire equation by 2:
  \(\mathrm{x}^2 - 3\mathrm{x} + \mathrm{y}^2 + \mathrm{y} = 22.5\)
• This makes the x² and y² coefficients equal to 1, which is required for standard form

3. SIMPLIFY by completing the square for x terms

• For \(\mathrm{x}^2 - 3\mathrm{x}\):
  • Take half the coefficient: \(-3 \div 2 = -1.5\)
  • Square it: \((-1.5)^2 = 2.25\)
  • So: \(\mathrm{x}^2 - 3\mathrm{x} = (\mathrm{x} - 1.5)^2 - 2.25\)

4. SIMPLIFY by completing the square for y terms

• For \(\mathrm{y}^2 + \mathrm{y}\):
  • Take half the coefficient: \(1 \div 2 = 0.5\)
  • Square it: \((0.5)^2 = 0.25\)
  • So: \(\mathrm{y}^2 + \mathrm{y} = (\mathrm{y} + 0.5)^2 - 0.25\)

5. SIMPLIFY by substituting and combining constants

• Substitute completed squares:
  \((\mathrm{x} - 1.5)^2 - 2.25 + (\mathrm{y} + 0.5)^2 - 0.25 = 22.5\)
• Move constants to right side:
  \((\mathrm{x} - 1.5)^2 + (\mathrm{y} + 0.5)^2 = 22.5 + 2.25 + 0.25 = 25\)
• Now in standard form: \((\mathrm{x} - 1.5)^2 + (\mathrm{y} + 0.5)^2 = 25\)

6. INFER the radius from standard form

• Since \((\mathrm{x} - \mathrm{h})^2 + (\mathrm{y} - \mathrm{k})^2 = \mathrm{r}^2\), we have \(\mathrm{r}^2 = 25\)
• Therefore: \(\mathrm{r} = \sqrt{25} = 5\)

Answer: A. 5

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY execution: Students make arithmetic errors when completing the square or combining constants

Many students correctly identify the need to complete the square but make calculation mistakes. Common errors include:

• Forgetting to square the half-coefficient when completing the square
• Sign errors when moving constants around
• Arithmetic mistakes when adding 22.5 + 2.25 + 0.25

This leads to an incorrect value for r² and selecting Choice B (6.5), Choice C (√40), or Choice D (√50).

Second Most Common Error:

Poor INFER reasoning: Students don't recognize the need to convert to standard form first

Some students try to find the radius directly from the general equation without converting to standard form. They might attempt to factor or use other approaches that don't lead to the radius.

This causes confusion and typically leads to guessing among the answer choices.

The Bottom Line:

This problem requires systematic algebraic manipulation skills. The key insight is recognizing that completing the square is the path to standard form, which directly reveals the radius.