If a = c + d, which of the following is equivalent to the expression x^2 - c^2 - 2cd...

1. TRANSLATE the problem information

• Given information:
  • \(\mathrm{a = c + d}\)
  • Expression to simplify: \(\mathrm{x^2 - c^2 - 2cd - d^2}\)
• Need to find which answer choice is equivalent to this expression

2. INFER the grouping strategy

• Look at the expression \(\mathrm{x^2 - c^2 - 2cd - d^2}\)
• The last three terms can be grouped: \(\mathrm{x^2 - (c^2 + 2cd + d^2)}\)
• This grouping might reveal a recognizable pattern

3. INFER the perfect square pattern

• Examine \(\mathrm{c^2 + 2cd + d^2}\)
• This matches the perfect square trinomial pattern: \(\mathrm{a^2 + 2ab + b^2 = (a + b)^2}\)
• Here: \(\mathrm{c^2 + 2cd + d^2 = (c + d)^2}\)
• So our expression becomes: \(\mathrm{x^2 - (c + d)^2}\)

4. INFER the substitution opportunity

• We know that \(\mathrm{a = c + d}\) from the given information
• Substitute: \(\mathrm{x^2 - (c + d)^2 = x^2 - a^2}\)

5. SIMPLIFY using difference of squares

• Recognize that \(\mathrm{x^2 - a^2}\) is a difference of squares
• Factor: \(\mathrm{x^2 - a^2 = (x + a)(x - a)}\)

Answer: C. \(\mathrm{(x + a)(x - a)}\)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students don't recognize the perfect square trinomial pattern in \(\mathrm{c^2 + 2cd + d^2}\)

Students might try to work with the original expression term by term without seeing the underlying structure. They may attempt to factor each term separately or try to match the expression directly to the answer choices without simplifying first. This leads to confusion and guessing among the answer choices.

Second Most Common Error:

Missing the substitution connection: Students recognize the perfect square but fail to connect \(\mathrm{a = c + d}\) to their work

Even if students successfully identify that \(\mathrm{c^2 + 2cd + d^2 = (c + d)^2}\), they might not realize they can substitute 'a' for \(\mathrm{(c + d)}\). They get stuck with \(\mathrm{x^2 - (c + d)^2}\) and can't see how to match this to any of the answer choices. This may lead them to select Choice D \(\mathrm{(x^2 - ax - a^2)}\) because it contains similar terms.

The Bottom Line:

This problem tests pattern recognition skills and the ability to see connections between given information and algebraic expressions. Success requires both recognizing the perfect square trinomial and making the strategic substitution.