Which of the following is equivalent to \(\mathrm{(a + \frac{b}{2})}\)?

1. INFER the approach needed

• We need to expand \((a + \frac{b}{2})^2\) to find its equivalent form
• This is a perfect square of a binomial, so we can use the formula \((x + y)^2 = x^2 + 2xy + y^2\)

2. SIMPLIFY by applying the perfect square formula

• Identify: \(x = a\) and \(y = \frac{b}{2}\)
• Apply formula: \((a + \frac{b}{2})^2 = a^2 + 2(a)(\frac{b}{2}) + (\frac{b}{2})^2\)
• SIMPLIFY each term:
  • First term: \(a^2\)
  • Middle term: \(2(a)(\frac{b}{2}) = ab\)
  • Last term: \((\frac{b}{2})^2 = \frac{b^2}{4}\)

3. SIMPLIFY to get the final expression

• Combine all terms: \(a^2 + ab + \frac{b^2}{4}\)
• Compare with answer choices

Answer: D. \(a^2 + ab + \frac{b^2}{4}\)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY execution: Students correctly set up the expansion but make arithmetic errors in the middle term or final term.

Common mistakes include:

• Computing \(2(a)(\frac{b}{2})\) as \(\frac{ab}{2}\) instead of \(ab\)
• Computing \((\frac{b}{2})^2\) as \(\frac{b^2}{2}\) instead of \(\frac{b^2}{4}\)
• Forgetting to include all three terms from the perfect square formula

This may lead them to select Choice B (\(a^2 + \frac{b^2}{4}\)) if they miss the middle term entirely, or Choice C (\(a^2 + \frac{ab}{2} + \frac{b^2}{2}\)) if they make both computational errors.

The Bottom Line:

This problem tests careful algebraic manipulation. Success requires systematic application of the perfect square formula with attention to fraction arithmetic, especially recognizing that \((\frac{b}{2})^2 = \frac{b^2}{4}\), not \(\frac{b^2}{2}\).