The curves y = x + 2 and y = x^2 - 6x + 8 intersect at two points in...

1. TRANSLATE the problem information

• Given information:
  • Curve 1: \(\mathrm{y = x + 2}\) (linear function)
  • Curve 2: \(\mathrm{y = x^2 - 6x + 8}\) (quadratic function)
  • Need: Sum of x-coordinates where curves intersect
• What this tells us: At intersection points, both curves have the same y-value for the same x-value.

2. INFER the solution approach

• Since both expressions equal y, we can set them equal to each other
• This will give us a quadratic equation to solve for the x-coordinates

3. TRANSLATE to mathematical equation

Set the expressions equal:

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

4. SIMPLIFY to standard quadratic form

• Move all terms to one side:
  \(\mathrm{x + 2 = x^2 - 6x + 8}\)
  \(\mathrm{0 = x^2 - 6x + 8 - x - 2}\)
  \(\mathrm{0 = x^2 - 7x + 6}\)
• Factor the quadratic:
  We need two numbers that multiply to 6 and add to -7
  Those numbers are -6 and -1
  \(\mathrm{0 = (x - 6)(x - 1)}\)

5. APPLY zero product property

• If \(\mathrm{(x - 6)(x - 1) = 0}\), then:
  \(\mathrm{x - 6 = 0}\) OR \(\mathrm{x - 1 = 0}\)
  \(\mathrm{x = 6}\) OR \(\mathrm{x = 1}\)

6. INFER the final answer

• The x-coordinates of intersection are 6 and 1
• \(\mathrm{Sum = 6 + 1 = 7}\)

Answer: C (7)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY execution: Students make algebraic errors when rearranging \(\mathrm{x + 2 = x^2 - 6x + 8}\) into standard form. Common mistakes include sign errors (getting \(\mathrm{x^2 - 5x + 6 = 0}\) instead of \(\mathrm{x^2 - 7x + 6 = 0}\)) or incorrect combining of like terms.

If they get \(\mathrm{x^2 - 5x + 6 = 0}\), this factors as \(\mathrm{(x - 3)(x - 2) = 0}\), giving \(\mathrm{x = 3}\) and \(\mathrm{x = 2}\), with \(\mathrm{sum = 5}\). Since 5 isn't an answer choice, this leads to confusion and guessing.

Second Most Common Error:

Poor INFER reasoning: Students correctly find the x-coordinates (6 and 1) but misunderstand what the question is asking for. They might calculate the actual intersection points by substituting back (getting \(\mathrm{(6,8)}\) and \(\mathrm{(1,3)}\)), then try to work with these coordinates instead of simply adding the x-values.

This confusion about what to calculate may lead them to select Choice A (1) if they focus on just the smaller x-coordinate, or cause them to get stuck and guess.

The Bottom Line:

This problem tests your ability to connect the geometric concept of intersection with algebraic equation-solving, then requires careful attention to what quantity the question actually wants you to find.