At how many points do the graphs of the equations y = x^2 - 5x + 6 and y =...

1. INFER what intersection means

• Given information:
  • First equation: \(\mathrm{y = x^2 - 5x + 6}\) (parabola)
  • Second equation: \(\mathrm{y = x - 2}\) (line)
• What this tells us: Intersection points occur where both equations give the same y-value for the same x-value

2. INFER the solution approach

• Since both expressions equal y, set them equal to each other
• This eliminates y and gives us an equation in terms of x only

3. SIMPLIFY by setting equations equal and rearranging

• Set up: \(\mathrm{x^2 - 5x + 6 = x - 2}\)
• Move all terms to one side: \(\mathrm{x^2 - 5x + 6 - x + 2 = 0}\)
• Combine like terms: \(\mathrm{x^2 - 6x + 8 = 0}\)

4. SIMPLIFY by factoring the quadratic

• Need two numbers that multiply to 8 and add to -6
• Those numbers are -2 and -4
• Factor: \(\mathrm{(x - 2)(x - 4) = 0}\)
• Solutions: \(\mathrm{x = 2}\) and \(\mathrm{x = 4}\)

5. INFER the final answer

• Since we found 2 distinct real solutions for x, the graphs intersect at exactly 2 points

Answer: C (2 points)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY execution: Algebraic mistakes when rearranging or combining like terms

Students might incorrectly rearrange \(\mathrm{x^2 - 5x + 6 = x - 2}\) and get something like:

• \(\mathrm{x^2 - 5x + 6 + x - 2 = 0}\) (wrong sign when moving terms)
• \(\mathrm{x^2 - 4x + 4 = 0}\) (incorrect combining of like terms)

These errors lead to different quadratic equations that factor differently, potentially giving 1 solution or no real solutions. This causes confusion and often leads to guessing among the answer choices.

Second Most Common Error:

Conceptual confusion about intersection: Not understanding that intersection means "where the graphs meet"

Some students might try to solve each equation separately or look for where one function equals zero, rather than where both functions have equal y-values. This leads them away from the systematic approach entirely and typically results in random answer selection.

The Bottom Line:

This problem tests whether students can connect the geometric concept of intersection with the algebraic technique of setting equations equal. The key insight is that intersection points satisfy both equations simultaneously.