The function g is defined by \(\mathrm{g(x) = x^2 + a(x - 2)}\), where a is a constant. If \(\mathrm{g(3)...

1. TRANSLATE the given information

• Given information:
  • Function: \(\mathrm{g(x) = x² + a(x - 2)}\) where 'a' is unknown
  • Condition: \(\mathrm{g(3) = 15}\)
  • Need to find: \(\mathrm{g(6)}\)

2. INFER the solution strategy

• Since 'a' is unknown but we need to find \(\mathrm{g(6)}\), we must find 'a' first
• The condition \(\mathrm{g(3) = 15}\) gives us the equation we need to solve for 'a'

3. TRANSLATE and SIMPLIFY to find 'a'

• Substitute \(\mathrm{x = 3}\) into the function:
  \(\mathrm{g(3) = (3)² + a(3 - 2)}\)
  \(\mathrm{g(3) = 9 + a(1)}\)
  \(\mathrm{g(3) = 9 + a}\)
• Since \(\mathrm{g(3) = 15}\):
  \(\mathrm{15 = 9 + a}\)
  \(\mathrm{a = 6}\)

4. SIMPLIFY by rewriting the function

• Now that \(\mathrm{a = 6}\), substitute back:
  \(\mathrm{g(x) = x² + 6(x - 2)}\)
  \(\mathrm{g(x) = x² + 6x - 12}\)

5. SIMPLIFY to find g(6)

• Substitute \(\mathrm{x = 6}\):
  \(\mathrm{g(6) = (6)² + 6(6) - 12}\)
  \(\mathrm{g(6) = 36 + 36 - 12}\)
  \(\mathrm{g(6) = 60}\)

Answer: D) 60

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students might try to find \(\mathrm{g(6)}\) directly without realizing they need to find 'a' first. They might attempt to substitute \(\mathrm{x = 6}\) into \(\mathrm{g(x) = x² + a(x - 2)}\) and get \(\mathrm{g(6) = 36 + 4a}\), then feel stuck because they don't know what 'a' equals.

This leads to confusion and guessing among the answer choices.

Second Most Common Error:

Poor SIMPLIFY execution: Students find \(\mathrm{a = 6}\) correctly but make algebraic errors when expanding \(\mathrm{g(x) = x² + 6(x - 2)}\). Common mistakes include:

• Forgetting to distribute the 6: getting \(\mathrm{g(x) = x² + 6x - 2}\) instead of \(\mathrm{g(x) = x² + 6x - 12}\)
• Sign errors when calculating \(\mathrm{g(6)}\), such as \(\mathrm{g(6) = 36 + 36 + 12 = 84}\)

These calculation errors may lead them to select other answer choices or feel uncertain about their work.

The Bottom Line:

This problem tests whether students recognize the logical sequence: use the given condition to find the unknown parameter first, then use that parameter to evaluate the function at the desired point. The key insight is that you can't evaluate \(\mathrm{g(6)}\) until you know what 'a' equals.