Which of the following expressions is equivalent to x^2 + 2xsqrt(3) + 3?

1. INFER the structure of the expression

• Given expression: \(\mathrm{x^2 + 2x\sqrt{3} + 3}\)
• This looks like it might be a perfect square trinomial in the form \(\mathrm{a^2 + 2ab + b^2}\)

2. INFER the pattern components

• For perfect square trinomials: \(\mathrm{a^2 + 2ab + b^2 = (a + b)^2}\)
• First term \(\mathrm{x^2}\) suggests \(\mathrm{a = x}\)
• Last term 3 suggests we need \(\mathrm{b^2 = 3}\), so \(\mathrm{b = \sqrt{3}}\)
• Check middle term: \(\mathrm{2ab = 2(x)(\sqrt{3}) = 2x\sqrt{3}}\) ✓

3. SIMPLIFY using the perfect square formula

• Since all three parts match the pattern \(\mathrm{a^2 + 2ab + b^2}\)
• We can write: \(\mathrm{x^2 + 2x\sqrt{3} + 3 = (x + \sqrt{3})^2}\)

Answer: C. \(\mathrm{(x + \sqrt{3})^2}\)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students don't recognize the perfect square trinomial pattern, especially with the radical term \(\mathrm{\sqrt{3}}\).

Many students see the \(\mathrm{2x\sqrt{3}}\) middle term and get confused because they're not comfortable working with radicals in factoring. They might try to expand the answer choices instead of recognizing the pattern, or they might incorrectly assume that since there's a \(\mathrm{\sqrt{3}}\), the answer must be more complex. This leads to confusion and guessing.

Second Most Common Error:

Inadequate SIMPLIFY execution: Students recognize it's a perfect square but incorrectly identify b.

They might see that the last term is 3 and incorrectly think \(\mathrm{b = 3}\) (instead of \(\mathrm{b = \sqrt{3}}\)), leading them to check if \(\mathrm{2x(3) = 6x}\) matches the middle term. When it doesn't, they get frustrated and may select Choice B: \(\mathrm{(x + 3)^2}\) based on the last term alone.

The Bottom Line:

This problem tests whether students can recognize perfect square patterns when radicals are involved. The key insight is that \(\mathrm{(\sqrt{3})^2 = 3}\), making the middle term verification crucial for confirming the pattern.