Question:In the xy-plane, the graph of y = x^2 intersects the line with equation y = 8x - 15 at...

1. TRANSLATE the intersection condition

• Given information:
  • Parabola: \(\mathrm{y = x^2}\)
  • Line: \(\mathrm{y = 8x - 15}\)
  • Need the larger x-coordinate of intersection points

• What this tells us: At intersection points, both graphs have the same y-value, so we can set their equations equal

2. TRANSLATE this insight into an equation

Set the right sides equal since both equal y:

\(\mathrm{x^2 = 8x - 15}\)

3. SIMPLIFY to solve the quadratic equation

Rearrange to standard form by moving all terms to one side:

\(\mathrm{x^2 - 8x + 15 = 0}\)

4. SIMPLIFY by factoring the quadratic

Look for two numbers that multiply to 15 and add to -8:

• Try -3 and -5: \(\mathrm{(-3) \times (-5) = 15}\) ✓ and \(\mathrm{(-3) + (-5) = -8}\) ✓

Factor: \(\mathrm{(x - 3)(x - 5) = 0}\)

5. INFER the solutions using zero product property

If \(\mathrm{(x - 3)(x - 5) = 0}\), then either:

• \(\mathrm{x - 3 = 0}\), so \(\mathrm{x = 3}\)
• \(\mathrm{x - 5 = 0}\), so \(\mathrm{x = 5}\)

6. APPLY CONSTRAINTS to identify q

Since \(\mathrm{p \lt q}\), and our solutions are 3 and 5:

• \(\mathrm{p = 3}\) (smaller value)
• \(\mathrm{q = 5}\) (larger value)

Answer: 5

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students may not recognize that "intersection points" means setting the equations equal to each other. Instead, they might try to solve each equation separately or attempt to graph without finding exact coordinates.

This leads to confusion and abandoning a systematic approach, causing them to guess randomly.

Second Most Common Error:

Poor SIMPLIFY execution: Students might make algebraic errors when rearranging \(\mathrm{x^2 = 8x - 15}\), such as incorrectly moving terms or struggling with factoring \(\mathrm{x^2 - 8x + 15 = 0}\). Some students may not recognize the factoring pattern or make sign errors.

This could lead them to get incorrect solutions or become stuck and resort to guessing.

Third Most Common Error:

Missing APPLY CONSTRAINTS reasoning: Students solve correctly to get \(\mathrm{x = 3}\) and \(\mathrm{x = 5}\), but then select \(\mathrm{p = 3}\) instead of \(\mathrm{q = 5}\), not carefully reading that the question asks for the larger x-coordinate.

This may lead them to answer 3 instead of the correct answer 5.

The Bottom Line:

This problem requires recognizing that intersection means equal y-values, competent algebraic manipulation of quadratic equations, and careful attention to which solution the question is asking for.