Which of the following expressions is equivalent to x^2 - 5?

1. INFER the algebraic pattern

• Looking at \(\mathrm{x^2 - 5}\), recognize this is in the form \(\mathrm{a^2 - b^2}\)
• Here: \(\mathrm{a = x}\) and \(\mathrm{b^2 = 5}\), so \(\mathrm{b = \sqrt{5}}\)
• This suggests we can use the difference of squares factorization

2. INFER the factorization approach

• The difference of squares pattern states: \(\mathrm{a^2 - b^2 = (a + b)(a - b)}\)
• For \(\mathrm{x^2 - 5}\): \(\mathrm{x^2 - (\sqrt{5})^2 = (x + \sqrt{5})(x - \sqrt{5})}\)
• Check if this matches any answer choices

3. SIMPLIFY to verify Choice C

• Choice C is \(\mathrm{(x + \sqrt{5})(x - \sqrt{5})}\)
• Using difference of squares: \(\mathrm{(x + \sqrt{5})(x - \sqrt{5}) = x^2 - (\sqrt{5})^2 = x^2 - 5}\) ✓
• This matches our original expression perfectly

4. SIMPLIFY other choices to confirm they're incorrect

• Choice A: \(\mathrm{(x + \sqrt{5})^2 = x^2 + 2x\sqrt{5} + 5 \neq x^2 - 5}\)
• Choice B: \(\mathrm{(x - \sqrt{5})^2 = x^2 - 2x\sqrt{5} + 5 \neq x^2 - 5}\)
• Choice D: \(\mathrm{(x + 5)(x - 1) = x^2 + 4x - 5 \neq x^2 - 5}\)

Answer: C

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students don't recognize the difference of squares pattern in \(\mathrm{x^2 - 5}\). Instead, they try to expand all the answer choices without a strategic approach, leading to confusion about which algebraic manipulations to perform. This often results in randomly guessing among the choices.

Second Most Common Error:

Poor SIMPLIFY execution: Students incorrectly expand perfect square binomials like \(\mathrm{(x + \sqrt{5})^2}\) or \(\mathrm{(x - \sqrt{5})^2}\), forgetting the middle term. They might think \(\mathrm{(x + \sqrt{5})^2 = x^2 + 5}\), making them incorrectly believe that Choice A or B could be equivalent to \(\mathrm{x^2 - 5}\). This may lead them to select Choice A or Choice B.

The Bottom Line:

This problem rewards students who can recognize algebraic patterns before diving into calculations. The key insight is seeing \(\mathrm{x^2 - 5}\) as a difference of squares, which immediately points to the correct factorization.