In the xy-plane, a line with equation y = x + 1 intersects a parabola with equation y = x^2...

1. INFER the intersection strategy

• Given information:
  • Line: \(\mathrm{y = x + 1}\)
  • Parabola: \(\mathrm{y = x^2 - 4x + 5}\)
  • Need: sum of x-coordinates where they intersect

• Key insight: Intersection points occur where both equations have the same x and y values, so I can set the y-expressions equal

2. Set up the equation

• Since both expressions equal y, set them equal to each other:

\(\mathrm{x + 1 = x^2 - 4x + 5}\)

3. SIMPLIFY to standard quadratic form

• Move all terms to one side:

\(\mathrm{0 = x^2 - 4x + 5 - x - 1}\)
\(\mathrm{0 = x^2 - 5x + 4}\)

4. INFER the most efficient approach

• I could factor or use the quadratic formula to find both roots, then add them
• More efficient: Use the sum of roots formula directly
• For \(\mathrm{ax^2 + bx + c = 0}\), sum of roots = \(\mathrm{-b/a}\)

5. Apply the sum formula

• In \(\mathrm{x^2 - 5x + 4 = 0}\): \(\mathrm{a = 1, b = -5, c = 4}\)
• Sum = \(\mathrm{-(-5)/1 = 5}\)

Answer: D (5)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students don't recognize that intersection points require setting the equations equal. Instead, they might try to solve each equation separately or get confused about what "intersection" means mathematically.

This leads to confusion and guessing among the answer choices.

Second Most Common Error:

Poor SIMPLIFY execution: Students make algebraic errors when rearranging to standard form, such as:

• Incorrectly combining like terms: writing \(\mathrm{x^2 - 4x - x}\) as \(\mathrm{x^2 - 3x}\) instead of \(\mathrm{x^2 - 5x}\)
• Sign errors when moving terms: getting \(\mathrm{x^2 - 5x - 4 = 0}\) instead of \(\mathrm{x^2 - 5x + 4 = 0}\)

These errors lead to incorrect coefficients, making the sum formula give wrong answers like Choice A (-5) or other incorrect values.

The Bottom Line:

This problem tests whether students understand the geometric meaning of intersection (equal y-values) and can efficiently use algebraic relationships (sum of roots) rather than brute-force calculation.