If \((\mathrm{x} + 5)^2 = 4\), which of the following is a possible value of x?

1. TRANSLATE the problem information

• Given: \((\mathrm{x} + 5)^2 = 4\)
• Find: Possible values of x from the given choices

2. CONSIDER ALL CASES when taking square roots

• When we have \((\mathrm{x} + 5)^2 = 4\), taking the square root of both sides gives us:
  \(\mathrm{x} + 5 = ±2\)
• This means we have TWO equations to solve:
  • \(\mathrm{x} + 5 = 2\) (positive case)
  • \(\mathrm{x} + 5 = -2\) (negative case)

3. SIMPLIFY each linear equation

• From \(\mathrm{x} + 5 = 2\):
  \(\mathrm{x} = 2 - 5\)
  \(\mathrm{x} = -3\)

• From \(\mathrm{x} + 5 = -2\):
  \(\mathrm{x} = -2 - 5\)
  \(\mathrm{x} = -7\)

4. Check which solutions appear in answer choices

• Our solutions are \(\mathrm{x} = -3\) and \(\mathrm{x} = -7\)
• Looking at choices: A. 1, B. -1, C. -2, D. -3
• Only \(\mathrm{x} = -3\) appears as choice D

Answer: D

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak CONSIDER ALL CASES skill: Students forget that taking the square root gives both positive and negative solutions. They might only solve \(\mathrm{x} + 5 = 2\), finding \(\mathrm{x} = -3\), but miss the \(\mathrm{x} + 5 = -2\) case entirely. While this still leads to the correct answer in this particular problem, it shows incomplete understanding that could cause errors in future problems.

Second Most Common Error:

Poor TRANSLATE reasoning: Students misread \((\mathrm{x} + 5)^2 = 4\) as \(\mathrm{x} + 5 = 4\), then solve to get \(\mathrm{x} = -1\). This leads them to select Choice B (-1).

Third Most Common Error:

TRANSLATE confusion about what to find: Students correctly find that \(\mathrm{x} + 5 = ±2\), but then give -2 as their final answer (confusing the value of \(\mathrm{x} + 5\) with the value of \(\mathrm{x}\)). This may lead them to select Choice C (-2).

The Bottom Line:

This problem tests whether students understand that square root operations produce two solutions and can distinguish between solving for an expression versus solving for the variable itself.