Question: The squared difference between 3x and 9 is 441. What is the positive value of x - 3?-73721

1. TRANSLATE the problem information

• Given information:
  • "The squared difference between 3x and 9 is 441"
  • Need to find the positive value of x - 3

• This translates to the equation: \((3\mathrm{x} - 9)^2 = 441\)

2. SIMPLIFY to isolate the linear expression

• Take the square root of both sides: \(3\mathrm{x} - 9 = ±\sqrt{441}\)
• Calculate: \(\sqrt{441} = 21\)
• So: \(3\mathrm{x} - 9 = ±21\)

3. CONSIDER ALL CASES from the ± solution

• Case 1: \(3\mathrm{x} - 9 = 21\)
  • Add 9: \(3\mathrm{x} = 30\)
  • Divide by 3: \(\mathrm{x} = 10\)

• Case 2: \(3\mathrm{x} - 9 = -21\)
  • Add 9: \(3\mathrm{x} = -12\)
  • Divide by 3: \(\mathrm{x} = -4\)

4. INFER what the question is actually asking for

• The question asks for \(\mathrm{x} - 3\), not just x
• Calculate \(\mathrm{x} - 3\) for both solutions:
  • When \(\mathrm{x} = 10\): \(\mathrm{x} - 3 = 10 - 3 = 7\)
  • When \(\mathrm{x} = -4\): \(\mathrm{x} - 3 = -4 - 3 = -7\)

5. APPLY CONSTRAINTS to select final answer

• The question specifically asks for the "positive value" of \(\mathrm{x} - 3\)
• Between 7 and -7, only 7 is positive

Answer: C) 7

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students may misinterpret "squared difference" as either \((3\mathrm{x})^2 - 9^2\) or as \(|3\mathrm{x} - 9|^2\) instead of \((3\mathrm{x} - 9)^2\).

This fundamental translation error leads to completely different equations and wrong solutions. Students might solve \((3\mathrm{x})^2 - 81 = 441\), getting \(3\mathrm{x} = ±\sqrt{522}\), which doesn't lead to any of the given answer choices. This causes confusion and random guessing.

Second Most Common Error:

Missing CONSIDER ALL CASES reasoning: Students correctly set up \((3\mathrm{x} - 9)^2 = 441\) and take the square root, but only consider the positive case: \(3\mathrm{x} - 9 = 21\), leading to \(\mathrm{x} = 10\) and \(\mathrm{x} - 3 = 7\).

While this actually gives the correct final answer by luck, they miss the complete mathematical reasoning. More problematically, some students might only consider the negative case and get \(\mathrm{x} = -4\), then calculate \(\mathrm{x} - 3 = -7\) and select Choice A (-7) without recognizing the question asks for the positive value.

The Bottom Line:

This problem tests whether students can accurately translate verbal descriptions into algebraic equations and then systematically work through all mathematical possibilities while applying the specific constraints given in the question.