If k = 3j, which of the following is equivalent to the expression x^2 - 9j^2?

1. TRANSLATE the given relationship

• Given information:
  • \(\mathrm{k = 3j}\)
  • Expression: \(\mathrm{x^2 - 9j^2}\)

• What this tells us: If \(\mathrm{k = 3j}\), then \(\mathrm{k^2 = (3j)^2 = 9j^2}\)

2. TRANSLATE the expression using substitution

• Since \(\mathrm{k^2 = 9j^2}\), we can rewrite:
  \(\mathrm{x^2 - 9j^2 = x^2 - k^2}\)

3. INFER the algebraic pattern

• Recognize that \(\mathrm{x^2 - k^2}\) is a difference of squares
• Apply the pattern: \(\mathrm{a^2 - b^2 = (a + b)(a - b)}\)
• Therefore: \(\mathrm{x^2 - k^2 = (x + k)(x - k)}\)

4. SIMPLIFY by checking answer choice (C)

• \(\mathrm{(x + k)(x - k)}\) expands to \(\mathrm{x^2 - k^2}\) ✓
• This matches our rewritten expression

Answer: C

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students don't make the connection that \(\mathrm{k = 3j}\) means \(\mathrm{k^2 = 9j^2}\). Instead, they work directly with the original expression \(\mathrm{x^2 - 9j^2}\) without recognizing the substitution opportunity.

Without this substitution, they might try to factor \(\mathrm{x^2 - 9j^2}\) directly, leading to confusion about how to handle the coefficient 9. They may incorrectly think this factors as \(\mathrm{(x + 3j)(x - 3j)}\), which would lead them to select Choice (D) \(\mathrm{(x - 3j)^2}\) since it contains similar terms.

Second Most Common Error:

Missing conceptual knowledge about difference of squares: Students might recognize that \(\mathrm{k^2 = 9j^2}\) but don't recall the difference of squares factoring pattern \(\mathrm{a^2 - b^2 = (a + b)(a - b)}\).

Without this pattern recognition, they get stuck trying to factor \(\mathrm{x^2 - k^2}\) and may guess or attempt to match it with answer choices that look familiar, like the perfect square patterns in choices (A), (B), or (D).

The Bottom Line:

This problem tests whether students can make strategic substitutions to reveal familiar algebraic patterns. The key insight is that algebraic expressions often become easier to work with when we substitute equivalent expressions that reveal standard patterns like difference of squares.