\((\frac{1}{2}\mathrm{x} + \frac{3}{2})(\frac{3}{2}\mathrm{x} + \frac{1}{2})\) The expression above is equivalent to ax^2 + bx + c, where a, b, and...

1. INFER the solution approach

• We need to expand \(\left(\frac{1}{2}\mathrm{x} + \frac{3}{2}\right)\left(\frac{3}{2}\mathrm{x} + \frac{1}{2}\right)\) to get it in the form \(\mathrm{ax}^2 + \mathrm{bx} + \mathrm{c}\)
• The best method is to use the distributive property (FOIL method)
• Once expanded, we'll combine like terms and identify the coefficient b

2. SIMPLIFY by expanding using FOIL

• First terms: \(\left(\frac{1}{2}\mathrm{x}\right)\left(\frac{3}{2}\mathrm{x}\right) = \left(\frac{1}{2}\right)\left(\frac{3}{2}\right)\mathrm{x}^2 = \frac{3}{4}\mathrm{x}^2\)
• Outer terms: \(\left(\frac{1}{2}\mathrm{x}\right)\left(\frac{1}{2}\right) = \left(\frac{1}{2}\right)\left(\frac{1}{2}\right)\mathrm{x} = \frac{1}{4}\mathrm{x}\)
• Inner terms: \(\left(\frac{3}{2}\right)\left(\frac{3}{2}\mathrm{x}\right) = \left(\frac{3}{2}\right)\left(\frac{3}{2}\right)\mathrm{x} = \frac{9}{4}\mathrm{x}\)
• Last terms: \(\left(\frac{3}{2}\right)\left(\frac{1}{2}\right) = \frac{3}{4}\)

Result so far: \(\frac{3}{4}\mathrm{x}^2 + \frac{1}{4}\mathrm{x} + \frac{9}{4}\mathrm{x} + \frac{3}{4}\)

3. SIMPLIFY by combining like terms

• Combine the x terms: \(\frac{1}{4}\mathrm{x} + \frac{9}{4}\mathrm{x} = \left(\frac{1}{4} + \frac{9}{4}\right)\mathrm{x} = \frac{10}{4}\mathrm{x} = \frac{5}{2}\mathrm{x}\)
• Final expression: \(\frac{3}{4}\mathrm{x}^2 + \frac{5}{2}\mathrm{x} + \frac{3}{4}\)

4. INFER the answer

• In the form \(\mathrm{ax}^2 + \mathrm{bx} + \mathrm{c}\), we have:
  • \(\mathrm{a} = \frac{3}{4}\)
  • \(\mathrm{b} = \frac{5}{2}\)
  • \(\mathrm{c} = \frac{3}{4}\)

Answer: \(\mathrm{b} = \frac{5}{2}\) (or 2.5)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY skill: Students make fraction multiplication errors during FOIL expansion or when combining like terms

For example, they might incorrectly calculate \(\left(\frac{3}{2}\right)\left(\frac{3}{2}\right) = \frac{6}{4}\) instead of \(\frac{9}{4}\), or make errors when adding \(\frac{1}{4} + \frac{9}{4}\). These arithmetic mistakes cascade through the problem, leading to an incorrect coefficient for the x term. This leads to confusion and potentially guessing among the answer choices.

Second Most Common Error:

Incomplete SIMPLIFY execution: Students expand correctly using FOIL but fail to combine like terms properly

They might leave their answer as \(\frac{3}{4}\mathrm{x}^2 + \frac{1}{4}\mathrm{x} + \frac{9}{4}\mathrm{x} + \frac{3}{4}\) without recognizing they need to add \(\frac{1}{4} + \frac{9}{4} = \frac{10}{4} = \frac{5}{2}\) for the x coefficient. This incomplete simplification prevents them from identifying the correct value of b.

The Bottom Line:

This problem tests your ability to work systematically with fraction arithmetic while applying the distributive property. The key is being methodical with each step of FOIL and careful when combining fractional coefficients.