\((\mathrm{x} - 1)^2 = -4\) How many distinct real solutions does the given equation have?...

1. INFER the fundamental property at work

• Look at the structure: \((\mathrm{x} - 1)^2 = -4\)
• Key insight: The left side is a perfect square
• Essential property: Any real number squared gives a non-negative result
• This means \((\mathrm{x} - 1)^2 \geq 0\) for any real value of x

2. INFER the contradiction

• Left side: \((\mathrm{x} - 1)^2 \geq 0\) (always non-negative)
• Right side: \(-4\) (negative)
• Contradiction: We need something that's both \(\geq 0\) and equal to \(-4\)
• This is mathematically impossible

3. APPLY CONSTRAINTS to reach the conclusion

• We're looking for real solutions only
• Since no real number can satisfy this equation, the number of distinct real solutions is zero

Answer: D. Zero

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students attempt to solve algebraically without recognizing the fundamental impossibility. They might try taking square roots: \((\mathrm{x} - 1) = \pm\sqrt{-4} = \pm 2\mathrm{i}\), then conclude there are "two solutions" without recognizing these are complex, not real.

This may lead them to select Choice B (Exactly two).

Second Most Common Error:

Missing conceptual knowledge: Students don't immediately recognize that squares of real numbers are always non-negative. They might attempt various algebraic manipulations, get confused by the negative right side, and abandon systematic solution.

This leads to confusion and guessing among the remaining choices.

The Bottom Line:

This problem tests whether students understand the fundamental properties of squares rather than algebraic manipulation skills. The key insight happens before any calculations - recognizing that the equation structure itself makes real solutions impossible.