A rectangle's area can be expressed as 2/3x^2 - 18. If this area can be factored in the form \(\frac{2}{3}(\mathrm{x}...

1. TRANSLATE the problem information

• Given: Rectangle area = \(\frac{2}{3}\mathrm{x}^2 - 18\)
• Need to factor in form: \(\frac{2}{3}(\mathrm{x} - \mathrm{m})(\mathrm{x} + \mathrm{m})\) where \(\mathrm{m} \gt 0\)
• Find: The value of m

2. INFER the solution strategy

• To find m, I need to expand the target form and match it to the given expression
• Since both expressions represent the same area, their coefficients must be equal

3. SIMPLIFY by expanding the target form

• \(\frac{2}{3}(\mathrm{x} - \mathrm{m})(\mathrm{x} + \mathrm{m}) = \frac{2}{3}(\mathrm{x}^2 - \mathrm{m}^2) = \frac{2}{3}\mathrm{x}^2 - \frac{2}{3}\mathrm{m}^2\)

4. INFER coefficient matching approach

• Set the expanded form equal to given expression:
  \(\frac{2}{3}\mathrm{x}^2 - \frac{2}{3}\mathrm{m}^2 = \frac{2}{3}\mathrm{x}^2 - 18\)
• The x² terms already match, so focus on constant terms

5. SIMPLIFY to solve for m

• Comparing constants: \(-\frac{2}{3}\mathrm{m}^2 = -18\)
• Multiply both sides by -1: \(\frac{2}{3}\mathrm{m}^2 = 18\)
• Multiply both sides by 3/2: \(\mathrm{m}^2 = 18 \times \frac{3}{2} = 27\)
• Take square root: \(\mathrm{m} = \sqrt{27} = \sqrt{9 \times 3} = 3\sqrt{3}\)

Answer: D \((3\sqrt{3})\)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY execution: Students make arithmetic errors when calculating \(\mathrm{m}^2 = 18 \times \frac{3}{2}\), getting \(\mathrm{m}^2 = 18\) instead of \(\mathrm{m}^2 = 27\).

When \(\mathrm{m}^2 = 18\), they calculate \(\mathrm{m} = \sqrt{18} = \sqrt{9 \times 2} = 3\sqrt{2}\).

This leads them to select Choice C \((3\sqrt{2})\).

Second Most Common Error:

Poor INFER strategy: Students attempt to factor out \(\frac{2}{3}\) first, writing \(\frac{2}{3}\mathrm{x}^2 - 18 = \frac{2}{3}(\mathrm{x}^2 - 27)\), then struggle to recognize this as a difference of squares.

Some incorrectly think \(\mathrm{x}^2 - 27 = (\mathrm{x} - 3)(\mathrm{x} + 9)\) or make other factoring errors, leading to confusion and guessing among the remaining choices.

The Bottom Line:

This problem tests whether students can work backwards from a desired factored form by expanding and equating coefficients. Success requires careful arithmetic and recognition that the difference of squares pattern drives the entire solution.