A line has equation y = x + 6. A parabola has equation y = x^2 - 6. What is...

1. INFER the approach for intersection points

• Given information:
  • Line: \(\mathrm{y = x + 6}\)
  • Parabola: \(\mathrm{y = x^2 - 6}\)
  • Need: positive x-coordinate where curves intersect

• Key insight: At intersection points, both equations give the same y-value for the same x-value, so we can set the right sides equal.

2. TRANSLATE the intersection condition into an equation

• Set the equations equal: \(\mathrm{x + 6 = x^2 - 6}\)

3. SIMPLIFY to standard quadratic form

• Move all terms to one side: \(\mathrm{0 = x^2 - x - 12}\)
• Now we have a quadratic equation to solve

4. SIMPLIFY by factoring the quadratic

• Need two numbers that multiply to -12 and add to -1
• Those numbers are -4 and 3: \(\mathrm{(-4) \times 3 = -12}\) and \(\mathrm{(-4) + 3 = -1}\)
• Factor: \(\mathrm{0 = (x - 4)(x + 3)}\)

5. APPLY zero product property and APPLY CONSTRAINTS

• From \(\mathrm{(x - 4)(x + 3) = 0}\), we get \(\mathrm{x = 4}\) or \(\mathrm{x = -3}\)
• Since we want the positive x-coordinate: \(\mathrm{x = 4}\)

Answer: 4

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students don't recognize that "intersection" means setting the equations equal to each other. Instead, they might try to graph both functions or substitute random values, leading to confusion and guessing.

Second Most Common Error:

Poor SIMPLIFY execution: Students correctly set up \(\mathrm{x + 6 = x^2 - 6}\) but make algebraic errors when rearranging to standard form, such as getting \(\mathrm{x^2 + x - 12 = 0}\) instead of \(\mathrm{x^2 - x - 12 = 0}\). This leads to incorrect factors and wrong solutions, causing them to get stuck and guess.

The Bottom Line:

This problem tests whether students understand the fundamental concept that intersection points occur when two functions have equal output values for the same input, requiring them to set expressions equal rather than trying to solve each equation separately.