Which expression is equivalent to 49x^4 - 196y^2?

1. INFER the pattern in the expression

• Looking at \(49\mathrm{x}^4 - 196\mathrm{y}^2\), I notice this is a subtraction of two terms
• Both terms appear to be perfect squares
• This suggests a difference of squares pattern: \(\mathrm{a}^2 - \mathrm{b}^2 = (\mathrm{a} - \mathrm{b})(\mathrm{a} + \mathrm{b})\)

2. SIMPLIFY to identify the values of a and b

• For the first term \(49\mathrm{x}^4\):
  • \(49 = 7^2\)
  • \(\mathrm{x}^4 = (\mathrm{x}^2)^2\)
  • So \(49\mathrm{x}^4 = (7\mathrm{x}^2)^2\), meaning \(\mathrm{a} = 7\mathrm{x}^2\)
• For the second term \(196\mathrm{y}^2\):
  • \(196 = 14^2\) (use calculator: \(\sqrt{196} = 14\))
  • \(\mathrm{y}^2 = (\mathrm{y})^2\)
  • So \(196\mathrm{y}^2 = (14\mathrm{y})^2\), meaning \(\mathrm{b} = 14\mathrm{y}\)

3. APPLY the difference of squares formula

• Using \(\mathrm{a}^2 - \mathrm{b}^2 = (\mathrm{a} - \mathrm{b})(\mathrm{a} + \mathrm{b})\)
• Substitute \(\mathrm{a} = 7\mathrm{x}^2\) and \(\mathrm{b} = 14\mathrm{y}\)
• Result: \((7\mathrm{x}^2 - 14\mathrm{y})(7\mathrm{x}^2 + 14\mathrm{y})\)

Answer: C. \((7\mathrm{x}^2 - 14\mathrm{y})(7\mathrm{x}^2 + 14\mathrm{y})\)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students incorrectly identify \(\mathrm{a} = 7\mathrm{x}\) instead of \(\mathrm{a} = 7\mathrm{x}^2\)

Students see \(49\mathrm{x}^4\) and think "the square root of \(\mathrm{x}^4\) is \(\mathrm{x}^2\)," but then mistakenly pair it with just the 7, getting \(\mathrm{a} = 7\mathrm{x}\). This leads them to factor as \((7\mathrm{x} - 14\mathrm{y})(7\mathrm{x} + 14\mathrm{y})\), which would equal \(49\mathrm{x}^2 - 196\mathrm{y}^2\), not the original \(49\mathrm{x}^4 - 196\mathrm{y}^2\).

This may lead them to select Choice A (\((7\mathrm{x} - 14\mathrm{y})(7\mathrm{x} + 14\mathrm{y})\)).

Second Most Common Error:

Missing conceptual knowledge: Students don't recognize the difference of squares pattern at all

Without knowing the \(\mathrm{a}^2 - \mathrm{b}^2 = (\mathrm{a} - \mathrm{b})(\mathrm{a} + \mathrm{b})\) formula, students might try other factoring methods or attempt to factor out common terms incorrectly. They might see the 49 and 196 and try to factor out 49, but this doesn't lead to a clean factorization.

This leads to confusion and guessing among the remaining choices.

The Bottom Line:

The key challenge is correctly identifying that \(\mathrm{x}^4 = (\mathrm{x}^2)^2\), not \((\mathrm{x})^2\). Students must recognize that when dealing with \(\mathrm{x}^4\) in a perfect square context, the "square root" gives \(\mathrm{x}^2\), not \(\mathrm{x}\).