Which of the following expressions is equivalent to \((\frac{1}{2}\mathrm{x} + 2)^2 - (\mathrm{x}^2 - 3\mathrm{x} + \frac{1}{2})\)?

1. INFER the solution strategy

• The expression has a squared binomial that needs expanding, followed by polynomial subtraction
• Approach: Expand first, then subtract, then combine like terms

2. SIMPLIFY by expanding the binomial

• Use the formula \((\mathrm{a} + \mathrm{b})^2 = \mathrm{a}^2 + 2\mathrm{ab} + \mathrm{b}^2\) where \(\mathrm{a} = \frac{1}{2}\mathrm{x}\) and \(\mathrm{b} = 2\):
  • \((\frac{1}{2}\mathrm{x})^2 = \frac{1}{4}\mathrm{x}^2\)
  • \(2(\frac{1}{2}\mathrm{x})(2) = 2\mathrm{x}\)
  • \(2^2 = 4\)
• So \((\frac{1}{2}\mathrm{x} + 2)^2 = \frac{1}{4}\mathrm{x}^2 + 2\mathrm{x} + 4\)

3. SIMPLIFY the subtraction by distributing the negative

• Original: \((\frac{1}{4}\mathrm{x}^2 + 2\mathrm{x} + 4) - (\mathrm{x}^2 - 3\mathrm{x} + \frac{1}{2})\)
• Distribute the negative: \((\frac{1}{4}\mathrm{x}^2 + 2\mathrm{x} + 4) + (-\mathrm{x}^2 + 3\mathrm{x} - \frac{1}{2})\)

4. SIMPLIFY by combining like terms

• \(\mathrm{x}^2\) terms: \(\frac{1}{4}\mathrm{x}^2 - \mathrm{x}^2 = \frac{1}{4}\mathrm{x}^2 - \frac{4}{4}\mathrm{x}^2 = -\frac{3}{4}\mathrm{x}^2\)
• \(\mathrm{x}\) terms: \(2\mathrm{x} + 3\mathrm{x} = 5\mathrm{x}\)
• Constants: \(4 - \frac{1}{2} = \frac{8}{2} - \frac{1}{2} = \frac{7}{2}\)

Answer: D. \(-\frac{3}{4}\mathrm{x}^2 + 5\mathrm{x} + \frac{7}{2}\)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY execution: Students make arithmetic errors with fractions, particularly when subtracting \(\mathrm{x}^2\) coefficients. They might incorrectly calculate \(\frac{1}{4} - 1\) as \(\frac{1}{4}\) or get confused converting \(1\) to \(\frac{4}{4}\).

This leads to getting positive \(\frac{1}{4}\mathrm{x}^2\) or \(\frac{5}{4}\mathrm{x}^2\) instead of the correct \(-\frac{3}{4}\mathrm{x}^2\), causing them to select Choice B (\(\frac{5}{4}\mathrm{x}^2 - \mathrm{x} + \frac{9}{2}\)) or Choice C (\(\frac{5}{4}\mathrm{x}^2 + 5\mathrm{x} + \frac{7}{2}\)).

Second Most Common Error:

Poor SIMPLIFY reasoning: Students incorrectly handle the constant terms when subtracting, getting \(4 - \frac{1}{2} = \frac{9}{2}\) instead of \(\frac{7}{2}\). They might subtract incorrectly or forget to convert \(4\) to eighths.

This may lead them to select Choice A (\(-\frac{3}{4}\mathrm{x}^2 + 5\mathrm{x} + \frac{9}{2}\)) with the wrong constant term.

The Bottom Line:

This problem tests systematic algebraic manipulation skills. Success requires careful fraction arithmetic and methodical combining of like terms - areas where small calculation errors can lead to wrong answer choices.