Question:Let p be a positive constant, and suppose |x| gt p. The value of x satisfies \((\mathrm{x}^2 - \mathrm{p}^2)^2 =...

1. TRANSLATE the problem information

• Given information:
  • p is a positive constant
  • \(|\mathrm{x}| \gt \mathrm{p}\) (x is farther from zero than p)
  • \((\mathrm{x}^2 - \mathrm{p}^2)^2 = 1600(\mathrm{x}^2 - \mathrm{p}^2)\)
  • Need to find which answer choice is a solution

2. INFER a strategic substitution

• The expression \((\mathrm{x}^2 - \mathrm{p}^2)\) appears twice in our equation
• Let \(\mathrm{y} = \mathrm{x}^2 - \mathrm{p}^2\) to simplify the equation
• This transforms our equation to: \(\mathrm{y}^2 = 1600\mathrm{y}\)

3. APPLY CONSTRAINTS to determine the sign of y

• Since \(|\mathrm{x}| \gt \mathrm{p}\), we know \(\mathrm{x}^2 \gt \mathrm{p}^2\)
• Therefore \(\mathrm{y} = \mathrm{x}^2 - \mathrm{p}^2 \gt 0\)
• This constraint will be crucial for selecting the correct solution

4. SIMPLIFY the quadratic equation

• Starting with \(\mathrm{y}^2 = 1600\mathrm{y}\)
• Rearrange: \(\mathrm{y}^2 - 1600\mathrm{y} = 0\)
• Factor: \(\mathrm{y}(\mathrm{y} - 1600) = 0\)
• Solutions: \(\mathrm{y} = 0\) or \(\mathrm{y} = 1600\)

5. APPLY CONSTRAINTS to select valid solution

• Since \(\mathrm{y} \gt 0\) (from step 3), we reject \(\mathrm{y} = 0\)
• Therefore \(\mathrm{y} = 1600\)

6. INFER the final step and SIMPLIFY

• Substitute back: \(\mathrm{x}^2 - \mathrm{p}^2 = 1600\)
• Solve for \(\mathrm{x}^2\): \(\mathrm{x}^2 = \mathrm{p}^2 + 1600\)
• Recognize that \(1600 = 40^2\): \(\mathrm{x}^2 = \mathrm{p}^2 + 40^2\)
• Take square root: \(\mathrm{x} = ±\sqrt{\mathrm{p}^2 + 40^2}\)

7. APPLY CONSTRAINTS to match answer choices

• Both \(\mathrm{x} = \sqrt{\mathrm{p}^2 + 40^2}\) and \(\mathrm{x} = -\sqrt{\mathrm{p}^2 + 40^2}\) are valid mathematically
• Looking at the choices, only \(-\sqrt{\mathrm{p}^2 + 40^2}\) appears

Answer: D

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students often get overwhelmed by the complex appearance of \((\mathrm{x}^2 - \mathrm{p}^2)^2 = 1600(\mathrm{x}^2 - \mathrm{p}^2)\) and try to expand everything instead of recognizing the substitution opportunity.

Without the substitution \(\mathrm{y} = \mathrm{x}^2 - \mathrm{p}^2\), they attempt to expand \((\mathrm{x}^2 - \mathrm{p}^2)^2\) and work with a quartic equation, leading to complicated algebra that's prone to errors. This often causes them to abandon systematic solution and guess.

Second Most Common Error:

Poor APPLY CONSTRAINTS reasoning: Students correctly find \(\mathrm{y} = 0\) or \(\mathrm{y} = 1600\) but forget to use the constraint \(|\mathrm{x}| \gt \mathrm{p}\), which means \(\mathrm{y} \gt 0\).

They accept \(\mathrm{y} = 0\), leading to \(\mathrm{x}^2 = \mathrm{p}^2\), so \(\mathrm{x} = ±\mathrm{p}\). Seeing \(-\mathrm{p}\) among the choices, they select Choice A (\(-\mathrm{p}\)) without recognizing this violates the original constraint \(|\mathrm{x}| \gt \mathrm{p}\).

The Bottom Line:

This problem tests whether students can recognize when substitution simplifies complex expressions and whether they remember to apply all given constraints throughout their solution.