\(\mathrm{g(x) = -\frac{1}{2}(x - 5)^2 + 8}\) The quadratic function g is defined as shown. In the xy-plane, the graph...

1. TRANSLATE the intersection condition

• Given information:
  • \(\mathrm{g(x) = -\frac{1}{2}(x - 5)^2 + 8}\)
  • Graph intersects horizontal line \(\mathrm{y = 6}\) at two points
  • Need to find the larger x-coordinate (n)

• What this tells us: We need to solve \(\mathrm{g(x) = 6}\)

2. TRANSLATE this into an algebraic equation

Set up the equation:

\(\mathrm{-\frac{1}{2}(x - 5)^2 + 8 = 6}\)

3. SIMPLIFY through systematic algebraic steps

• Subtract 8 from both sides:

\(\mathrm{-\frac{1}{2}(x - 5)^2 = -2}\)

• Multiply both sides by -2:

\(\mathrm{(x - 5)^2 = 4}\)

• Take the square root of both sides:

\(\mathrm{x - 5 = ±2}\)

4. CONSIDER ALL CASES from the square root

• Solve both cases:
  • \(\mathrm{x - 5 = 2}\) → \(\mathrm{x = 7}\)
  • \(\mathrm{x - 5 = -2}\) → \(\mathrm{x = 3}\)

5. APPLY CONSTRAINTS to identify the answer

• Since \(\mathrm{m \lt n}\) and we found \(\mathrm{x = 3}\) and \(\mathrm{x = 7}\):
  • \(\mathrm{m = 3}\) (smaller value)
  • \(\mathrm{n = 7}\) (larger value)

Answer: D (7)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak CONSIDER ALL CASES reasoning: Students take the square root but forget about the ± symbol, only considering one solution.

When they solve \(\mathrm{(x - 5)^2 = 4}\), they might only think \(\mathrm{x - 5 = 2}\), giving them \(\mathrm{x = 7}\). They miss the negative case entirely and don't realize there should be two intersection points. This leads to confusion about which value represents n since they only found one solution, causing them to second-guess their work and potentially guess incorrectly.

Second Most Common Error:

Poor SIMPLIFY execution: Students make algebraic mistakes during the manipulation steps.

Common errors include incorrect sign handling when multiplying by -2, or arithmetic mistakes when isolating the squared term. For instance, they might get \(\mathrm{(x - 5)^2 = -4}\) instead of \(\mathrm{(x - 5)^2 = 4}\), leading them to conclude there are no real solutions. This may lead them to select Choice B (5) by incorrectly thinking the vertex x-coordinate is the answer.

The Bottom Line:

This problem tests whether students can systematically solve quadratic equations while remembering that square roots produce two solutions. The vertex form makes the algebra manageable, but students must stay organized through multiple algebraic steps and remember to consider both roots.