A physics student is analyzing motion where the equation \((\mathrm{x} - 7)^2 = \mathrm{k} - 18\) represents a relationship between...

1. INFER the key constraint for real solutions

• Given equation: \((\mathrm{x} - 7)^2 = \mathrm{k} - 18\)
• Key insight: The left side \((\mathrm{x} - 7)^2\) is always non-negative since it's a square
• For the equation to have real solutions, both sides must be able to equal each other

2. INFER the mathematical requirement

• Since \((\mathrm{x} - 7)^2 \geq 0\) for all real x, we need: \(\mathrm{k} - 18 \geq 0\)
• This ensures the right side can match the non-negative left side

3. SIMPLIFY the inequality

• From \(\mathrm{k} - 18 \geq 0\)
• Add 18 to both sides: \(\mathrm{k} \geq 18\)

4. INFER when no real solutions exist

• Real solutions exist when \(\mathrm{k} \geq 18\)
• No real solutions exist when \(\mathrm{k} \lt 18\)
• Therefore, \(\mathrm{m} = 18\)

Answer: D) 18

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students don't recognize that \((\mathrm{x} - 7)^2\) must always be non-negative, so they attempt to solve the equation algebraically without considering the constraint on k.

They might try expanding \((\mathrm{x} - 7)^2 = \mathrm{x}^2 - 14\mathrm{x} + 49\) and set it equal to \(\mathrm{k} - 18\), leading to confusion about when solutions exist. This leads to guessing among the answer choices.

Second Most Common Error:

Conceptual confusion about real solutions: Students might think "no real solutions" occurs when \(\mathrm{k} = 18\) (the boundary case) rather than when \(\mathrm{k} \lt 18\).

This misunderstanding of the inequality direction may lead them to select Choice B (7) or guess incorrectly.

The Bottom Line:

This problem requires recognizing that geometric/algebraic constraints (squares being non-negative) create conditions on parameters for solutions to exist. Students must connect the algebraic form to the fundamental properties of real numbers.