Question:If \((y - 3)^2 - 16 = 0\), which of the following could be the value of y?-1152

1. INFER the best approach

• Given equation: \(\mathrm{(y - 3)^2 - 16 = 0}\)
• Since we have a perfect square term, the most efficient strategy is to isolate it first
• This will create a simpler equation to solve

2. SIMPLIFY by isolating the squared term

• Add 16 to both sides: \(\mathrm{(y - 3)^2 - 16 + 16 = 0 + 16}\)
• Result: \(\mathrm{(y - 3)^2 = 16}\)

3. CONSIDER ALL CASES when taking the square root

• Take the square root of both sides: \(\mathrm{\sqrt{(y - 3)^2} = \sqrt{16}}\)
• Remember: when we take the square root of both sides, we get ±: \(\mathrm{y - 3 = ±4}\)
• This gives us two separate equations to solve

4. SIMPLIFY both equations

• Case 1: \(\mathrm{y - 3 = 4}\)
  • Add 3 to both sides: \(\mathrm{y = 7}\)
• Case 2: \(\mathrm{y - 3 = -4}\)
  • Add 3 to both sides: \(\mathrm{y = -1}\)

5. Check which solution appears in the answer choices

• Our solutions: \(\mathrm{y = 7}\) or \(\mathrm{y = -1}\)
• Looking at choices: (A) -1, (B) 1, (C) 5, (D) 2
• Choice (A) matches \(\mathrm{y = -1}\)

Answer: (A) -1

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak CONSIDER ALL CASES skill: Students take the square root but forget about the ± sign, only solving \(\mathrm{y - 3 = 4}\) to get \(\mathrm{y = 7}\). Since 7 doesn't appear in the answer choices, they get confused and either guess randomly or assume they made an error somewhere else in their work.

Second Most Common Error:

Poor SIMPLIFY execution: Students correctly set up \(\mathrm{y - 3 = -4}\) but make a sign error when solving, calculating \(\mathrm{y = -4 - 3 = -7}\) instead of \(\mathrm{y = -4 + 3 = -1}\). Since -7 isn't among the choices, this leads to confusion and guessing.

The Bottom Line:

This problem tests whether students remember that square root operations create two solutions, not just one. The key insight is that even though only one of the two solutions appears in the answer choices, recognizing that both solutions exist is essential for confidently selecting the correct answer.