Question:m^2/(sqrt(m^2-n^2)) - n^2/(sqrt(m^2-n^2)) = 18In the given equation, n is a positive constant. Which of the following is one of...

1. TRANSLATE the problem information

• Given equation: \(\frac{\mathrm{m}^2}{\sqrt{\mathrm{m}^2-\mathrm{n}^2}} - \frac{\mathrm{n}^2}{\sqrt{\mathrm{m}^2-\mathrm{n}^2}} = 18\)
• We need to find one solution for m
• n is a positive constant

2. SIMPLIFY by factoring the common denominator

• Notice both terms have the same denominator: \(\sqrt{\mathrm{m}^2-\mathrm{n}^2}\)
• Factor it out: \(\frac{\mathrm{m}^2 - \mathrm{n}^2}{\sqrt{\mathrm{m}^2-\mathrm{n}^2}} = 18\)
• This is the crucial simplification that makes the problem workable

3. SIMPLIFY the left side further

• We have \(\mathrm{m}^2 - \mathrm{n}^2\) divided by \(\sqrt{\mathrm{m}^2 - \mathrm{n}^2}\)
• This simplifies to: \(\sqrt{\mathrm{m}^2-\mathrm{n}^2} = 18\)
• Key insight: When you divide an expression by its square root, you get the square root

4. INFER the next strategic step

• To eliminate the square root, square both sides
• \(\sqrt{\mathrm{m}^2-\mathrm{n}^2} = 18\) becomes \(\mathrm{m}^2 - \mathrm{n}^2 = 324\)

5. SIMPLIFY to solve for m

• From \(\mathrm{m}^2 - \mathrm{n}^2 = 324\), add \(\mathrm{n}^2\) to both sides
• \(\mathrm{m}^2 = \mathrm{n}^2 + 324\)
• Take the square root: \(\mathrm{m} = \pm\sqrt{\mathrm{n}^2 + 324}\)

6. CONSIDER ALL CASES for the final answer

• We get two solutions: \(\mathrm{m} = \sqrt{\mathrm{n}^2 + 324}\) and \(\mathrm{m} = -\sqrt{\mathrm{n}^2 + 324}\)
• The question asks for "one of the solutions"
• The negative solution \(\mathrm{m} = -\sqrt{\mathrm{n}^2 + 324}\) matches answer choice D

Answer: D

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY skill: Students often struggle with the key simplification step where \(\frac{\mathrm{m}^2 - \mathrm{n}^2}{\sqrt{\mathrm{m}^2-\mathrm{n}^2}}\) becomes \(\sqrt{\mathrm{m}^2-\mathrm{n}^2}\). They might try to work with the original complex fraction form, leading to algebraic confusion and making the problem much more difficult than necessary. This leads to abandoning systematic solution and guessing.

Second Most Common Error:

Poor CONSIDER ALL CASES execution: Students correctly work through the algebra but forget that taking the square root yields both positive and negative solutions. They might only consider \(\mathrm{m} = \sqrt{\mathrm{n}^2 + 324}\) and not find it among the answer choices, leading to confusion and incorrect selection of a different choice.

The Bottom Line:

This problem tests your ability to recognize that complex rational expressions can often be simplified dramatically. The key breakthrough is seeing that a fraction with matching expressions in numerator and denominator (one squared, one square-rooted) simplifies to just the square root.