The distance between the point \(\mathrm{(x, 0)}\) and the point \(\mathrm{(0, 4)}\) is 5 units. What is the negative value...

1. TRANSLATE the problem information

• Given information:
  • Point 1: \(\mathrm{(x, 0)}\)
  • Point 2: \(\mathrm{(0, 4)}\)
  • Distance between them: 5 units
• What we need: The negative value of x

2. TRANSLATE the distance relationship into mathematical form

• The distance formula between two points \(\mathrm{(x_1, y_1)}\) and \(\mathrm{(x_2, y_2)}\) is:
  \(\mathrm{d} = \sqrt{(\mathrm{x_2}-\mathrm{x_1})^2 + (\mathrm{y_2}-\mathrm{y_1})^2}\)
• Substituting our points \(\mathrm{(x, 0)}\) and \(\mathrm{(0, 4)}\):
  \(5 = \sqrt{(0-\mathrm{x})^2 + (4-0)^2}\)
  \(5 = \sqrt{\mathrm{x}^2 + 16}\)

3. SIMPLIFY to solve for x

• Square both sides to eliminate the square root:
  \(25 = \mathrm{x}^2 + 16\)
• Subtract 16 from both sides:
  \(\mathrm{x}^2 = 9\)
• Take the square root of both sides:
  \(\mathrm{x} = ±3\)

4. CONSIDER ALL CASES to find the correct answer

• We have two solutions: \(\mathrm{x} = 3\) and \(\mathrm{x} = -3\)
• The problem specifically asks for the negative value
• Therefore: \(\mathrm{x} = -3\)

Answer: C) -3

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students may struggle to properly set up the distance formula, especially with the coordinate substitution. They might write \(\sqrt{(\mathrm{x}-0)^2 + (0-4)^2}\) incorrectly as \(\sqrt{(\mathrm{x}-0)^2 + (4-0)^2}\) or make sign errors when expanding \((-\mathrm{x})^2\) and \((-4)^2\).

This leads to calculation errors that produce wrong x-values, causing them to select an incorrect answer choice or get confused and guess.

Second Most Common Error:

Poor CONSIDER ALL CASES execution: Students correctly solve to get \(\mathrm{x}^2 = 9\) and find \(\mathrm{x} = ±3\), but then either forget the problem asks specifically for the negative value, or they assume the positive value is always the "correct" answer in geometry problems.

This may lead them to select Choice A (-5) if they made calculation errors, or get confused about which sign to choose.

The Bottom Line:

This problem tests whether students can accurately translate a geometric distance relationship into algebraic form and then carefully track the signs through their solution. Success requires both solid setup skills and attention to the specific requirement for the negative solution.