If x^2 = a + b and y^2 = a + c, which of the following is equal to \((\mathrm{x}^2...

1. TRANSLATE the given information

• Given information:
  • \(\mathrm{x^2 = a + b}\)
  • \(\mathrm{y^2 = a + c}\)
• We need to find \(\mathrm{(x^2 - y^2)^2}\)

2. INFER the most efficient approach

• Rather than immediately expanding \(\mathrm{(x^2 - y^2)^2}\), let's first simplify what's inside the parentheses
• This will make the squaring step much cleaner

3. SIMPLIFY \(\mathrm{x^2 - y^2}\) by substitution

• \(\mathrm{x^2 - y^2 = (a + b) - (a + c)}\)
• \(\mathrm{x^2 - y^2 = a + b - a - c}\)
• \(\mathrm{x^2 - y^2 = b - c}\)

4. SIMPLIFY by squaring the result

• \(\mathrm{(x^2 - y^2)^2 = (b - c)^2}\)
• Using the expansion \(\mathrm{(u - v)^2 = u^2 - 2uv + v^2}\):
• \(\mathrm{(b - c)^2 = b^2 - 2bc + c^2}\)

Answer: B. \(\mathrm{b^2 - 2bc + c^2}\)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY execution: Students attempt to expand \(\mathrm{(x^2 - y^2)^2}\) directly using the formula \(\mathrm{(u - v)^2 = u^2 - 2uv + v^2}\), treating \(\mathrm{x^2}\) and \(\mathrm{y^2}\) as single terms without first substituting their values.

They might write \(\mathrm{(x^2 - y^2)^2 = (x^2)^2 - 2(x^2)(y^2) + (y^2)^2}\), then get confused about how to handle \(\mathrm{x^4}\) and \(\mathrm{y^4}\) terms, or make errors when substituting and expanding the complex expressions. This leads to calculation mistakes and incorrect combinations of terms.

This may lead them to select Choice A (\(\mathrm{a^2 - 2ac + c^2}\)) or get confused and guess.

Second Most Common Error:

Poor SIMPLIFY reasoning: Students correctly substitute to get \(\mathrm{(a + b - a - c)^2 = (b - c)^2}\), but then make sign errors when expanding, particularly with the middle term.

They might write \(\mathrm{(b - c)^2 = b^2 + 2bc + c^2}\) (using the wrong sign for the middle term) or \(\mathrm{b^2 - 2bc - c^2}\) (wrong sign on the last term). These algebraic expansion errors are common when students rush through the squaring process.

This causes them to get stuck and guess among the remaining choices.

The Bottom Line:

This problem tests whether students can efficiently organize their algebraic work and execute expansions accurately. The key insight is recognizing that simplifying the expression inside the parentheses first makes the problem much more manageable than trying to expand everything at once.