Consider the system of equations where y = x^2 - 6x + 8 and y = 2x - 4. If...

1. TRANSLATE the system into a solution strategy

• Given information:
  • \(\mathrm{y = x^2 - 6x + 8}\) (parabola)
  • \(\mathrm{y = 2x - 4}\) (line)
  • Need solution \(\mathrm{(a, b)}\) where \(\mathrm{a \gt 2}\)
• Since both expressions equal y, we can set them equal to each other to find intersection points

2. SIMPLIFY by setting the equations equal and rearranging

• Set the right sides equal:
  \(\mathrm{x^2 - 6x + 8 = 2x - 4}\)
• Move all terms to one side:
  \(\mathrm{x^2 - 6x + 8 - 2x + 4 = 0}\)
  \(\mathrm{x^2 - 8x + 12 = 0}\)

3. SIMPLIFY by factoring the quadratic

• Look for two numbers that multiply to 12 and add to -8
• Those numbers are -2 and -6: \(\mathrm{(-2)(-6) = 12}\), \(\mathrm{(-2) + (-6) = -8}\)
• Factor: \(\mathrm{(x - 2)(x - 6) = 0}\)
• Solutions: \(\mathrm{x = 2}\) or \(\mathrm{x = 6}\)

4. APPLY CONSTRAINTS to select the final answer

• Since we need \(\mathrm{a \gt 2}\), we select \(\mathrm{a = 6}\)
• We can verify: When \(\mathrm{x = 6}\), both equations give \(\mathrm{y = 8}\)

Answer: 6

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students may not recognize that 'system of equations' means finding where the two functions intersect, so they should set the expressions equal to each other.

Instead, they might try to solve each equation individually or attempt to use substitution/elimination methods that don't apply directly here. This leads to confusion and guessing rather than systematic solution.

Second Most Common Error:

Poor SIMPLIFY execution: Students make algebraic errors when combining like terms or factoring the quadratic.

Common mistakes include:

• Incorrectly combining \(\mathrm{-6x - 2x}\) to get \(\mathrm{-4x}\) instead of \(\mathrm{-8x}\)
• Struggling to factor \(\mathrm{x^2 - 8x + 12}\), especially finding the correct factor pair
• Sign errors when moving terms across the equals sign

This may lead them to get incorrect x-values or become stuck partway through the problem.

The Bottom Line:

This problem tests whether students understand that solving a system means finding intersection points, and it requires solid algebraic manipulation skills. The constraint \(\mathrm{a \gt 2}\) is the final filter that separates students who solve completely from those who stop after finding any solution.