If \(\mathrm{k = ((m/4) - 1)^2}\) and m gt 4, what is m in terms of k?

1. TRANSLATE the problem information

• Given: \(\mathrm{k = ((m/4) - 1)^2}\) and \(\mathrm{m \gt 4}\)
• Find: m in terms of k

2. INFER the solving strategy

• Since we have k equal to a squared expression, we need to "undo" the square by taking square root
• The constraint \(\mathrm{m \gt 4}\) will be crucial for handling the square root properly

3. SIMPLIFY by taking the square root of both sides

• Starting with: \(\mathrm{k = ((m/4) - 1)^2}\)
• Take square root: \(\mathrm{\sqrt{k} = |((m/4) - 1)|}\)

4. INFER how to handle the absolute value

• Since \(\mathrm{m \gt 4}\), we have \(\mathrm{m/4 \gt 1}\)
• This means \(\mathrm{(m/4) - 1 \gt 0}\)
• When the expression inside absolute value is positive, \(\mathrm{|x| = x}\)
• Therefore: \(\mathrm{\sqrt{k} = (m/4) - 1}\)

5. SIMPLIFY to isolate m

• Add 1 to both sides: \(\mathrm{\sqrt{k} + 1 = m/4}\)
• Multiply both sides by 4: \(\mathrm{m = 4(\sqrt{k} + 1)}\)
• Distribute: \(\mathrm{m = 4\sqrt{k} + 4}\)

Answer: A) \(\mathrm{m = 4\sqrt{k} + 4}\)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students don't use the constraint \(\mathrm{m \gt 4}\) to determine that \(\mathrm{(m/4) - 1}\) is positive, so they get confused about how to handle the absolute value after taking the square root. They might keep the ± symbol or make sign errors, leading to incorrect algebraic manipulation. This leads to confusion and guessing among the answer choices.

Second Most Common Error:

Poor SIMPLIFY execution: Students correctly get to \(\mathrm{\sqrt{k} + 1 = m/4}\) but make errors in the final step. They might forget to multiply the entire right side by 4, getting \(\mathrm{m = \sqrt{k} + 4}\), or incorrectly distribute the multiplication. This may lead them to select Choice C (\(\mathrm{\sqrt{k} + 4}\)) or Choice B (\(\mathrm{2\sqrt{k} + 4}\)).

The Bottom Line:

This problem tests whether students can systematically work backwards from a squared expression while properly using given constraints. The key insight is recognizing that \(\mathrm{m \gt 4}\) eliminates ambiguity about signs when taking square roots.