x^2 = 64 Which of the following values of x satisfies the given equation?...

1. SIMPLIFY the equation by taking square roots

• Given: \(\mathrm{x^2 = 64}\)
• Take the square root of both sides: \(\mathrm{x = \pm\sqrt{64}}\)
• Calculate: \(\mathrm{x = \pm8}\)

2. CONSIDER ALL CASES for the complete solution

• This gives us two solutions:
  • \(\mathrm{x = 8}\) (positive solution)
  • \(\mathrm{x = -8}\) (negative solution)

3. Match with answer choices

• Looking at the given choices:
  • A. -8 ← This matches our negative solution
  • B. 4
  • C. 32
  • D. 128
• Only -8 appears among the choices

Answer: A. -8

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak CONSIDER ALL CASES skill: Students often forget that square root operations produce both positive and negative results.

They correctly take the square root of both sides but only write \(\mathrm{x = 8}\), missing the negative solution \(\mathrm{x = -8}\). Since 8 isn't among the answer choices, this leads to confusion and guessing between the available options.

Second Most Common Error:

Poor SIMPLIFY execution: Students make computational errors with basic operations instead of taking square roots properly.

For example, they might think "what divided by 2 gives 64?" leading to 32, or "what times 2 gives 64?" leading to 128. This may lead them to select Choice C (32) or Choice D (128).

The Bottom Line:

The key insight is remembering that quadratic equations typically have two solutions, and square root operations always involve both positive and negative possibilities. Don't stop at just the positive answer!