\((\mathrm{x} - 3)^2 + (\mathrm{y} + 4)^2 = -25\) How many distinct real ordered pairs \((\mathrm{x}, \mathrm{y})\) satisfy the given...

1. TRANSLATE the problem information

• Given equation: \((\mathrm{x} - 3)^2 + (\mathrm{y} + 4)^2 = -25\)
• Need to find: How many distinct real ordered pairs \((\mathrm{x}, \mathrm{y})\) satisfy this equation
• Answer choices suggest we might have zero, one, two, or infinitely many solutions

2. INFER the key mathematical insight

• The left side contains two squared terms: \((\mathrm{x} - 3)^2\) and \((\mathrm{y} + 4)^2\)
• Any squared expression with real numbers is always non-negative
• This means: \((\mathrm{x} - 3)^2 \geq 0\) and \((\mathrm{y} + 4)^2 \geq 0\) for all real x and y
• Therefore: \((\mathrm{x} - 3)^2 + (\mathrm{y} + 4)^2 \geq 0\) for all real x and y

3. APPLY CONSTRAINTS to determine solution existence

• Left side: \((\mathrm{x} - 3)^2 + (\mathrm{y} + 4)^2 \geq 0\) (always non-negative)
• Right side: \(-25\) (negative)
• Since a non-negative number cannot equal a negative number, this equation has no real solutions

Answer: (D) Zero

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students may try to solve the equation algebraically without recognizing the fundamental impossibility. They might attempt to expand the squares or use substitution methods, getting bogged down in calculations without stepping back to see the big picture. This leads to confusion and guessing among the answer choices.

Second Most Common Error:

Missing conceptual knowledge about squared terms: Students may not immediately recall that squared expressions are always non-negative for real numbers. Without this key insight, they might think the equation could have solutions and attempt various algebraic approaches. This may lead them to select Choice (C) (Infinitely many) if they incorrectly think this represents a circle or other familiar shape.

The Bottom Line:

This problem tests conceptual understanding more than computational skill. The key insight is recognizing when an equation is impossible to solve based on the fundamental properties of squared expressions, rather than attempting complex algebraic manipulations.