The circle with center at \(\left(-\frac{3}{2}, -\frac{1}{2}\right)\) and radius 7 is represented by the equation x^2 + 3x + y^2...

1. TRANSLATE the problem information

• Given information:
  • Circle center: \((-3/2, -1/2)\)
  • Radius: \(7\)
  • Target equation form: \(x² + 3x + y² + y = c\)
• What this tells us: We need to start with standard form and convert to the given format

2. INFER the approach

• Since we know center and radius, write the standard circle equation first
• Then expand it algebraically to match the target form
• The value of c will be whatever's left after moving constant terms to the right side

3. TRANSLATE center and radius into standard form

• Standard form: \((x - h)² + (y - k)² = r²\)
• With center \((-3/2, -1/2)\): \((x - (-3/2))² + (y - (-1/2))² = 7²\)
• This gives us: \((x + 3/2)² + (y + 1/2)² = 49\)

4. SIMPLIFY by expanding the squared expressions

• Expand \((x + 3/2)²\):
  \((x + 3/2)² = x² + 2x(3/2) + (3/2)²\)
  \(= x² + 3x + 9/4\)

• Expand \((y + 1/2)²\):
  \((y + 1/2)² = y² + 2y(1/2) + (1/2)²\)
  \(= y² + y + 1/4\)

5. SIMPLIFY by combining and rearranging terms

• Full expanded equation: \(x² + 3x + 9/4 + y² + y + 1/4 = 49\)
• Rearrange to match target form: \(x² + 3x + y² + y = 49 - 9/4 - 1/4\)
• Combine the fractions: \(49 - 9/4 - 1/4 = 49 - 10/4 = 49 - 5/2\)

6. SIMPLIFY the final calculation

• Convert 49 to halves: \(49 = 98/2\)
• Final calculation: \(98/2 - 5/2 = 93/2\)

Answer: 93/2

Alternative acceptable forms: 46.5, 46 1/2

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY skill: Making sign errors when expanding \((x + 3/2)²\) or \((y + 1/2)²\)

Students might incorrectly expand \((x + 3/2)²\) as \(x² + 3x + 9/4\) but forget the middle term coefficient comes from \(2x(3/2) = 3x\), or they might write \(x² + 6x + 9/4\). Similarly with the y-term expansion. These algebraic errors cascade through the rest of the problem, leading to incorrect values of c.

Second Most Common Error:

Poor SIMPLIFY execution: Fraction arithmetic mistakes when combining \(9/4 + 1/4\) or subtracting from 49

Students often struggle with the final calculation \(49 - 9/4 - 1/4\), either making errors like \(49 - 9/4 - 1/4 = 49 - 8/4 = 49 - 2 = 47\), or getting confused about converting between whole numbers and fractions. This leads to various incorrect c values.

The Bottom Line:

This problem requires solid algebraic manipulation skills and careful fraction arithmetic. The conceptual understanding (standard circle form) is straightforward, but the execution demands precision in multiple algebraic steps.