Question:In the xy-plane, the line with equation y = 4x + 1 intersects the parabola with equation y = x^2...

1. TRANSLATE the problem information

• Given information:
  • Line: \(\mathrm{y = 4x + 1}\)
  • Parabola: \(\mathrm{y = x^2 + px + 5}\)
  • They intersect at exactly one point
  • p is positive

2. INFER what "exactly one intersection point" means mathematically

• When we set the equations equal, we get a quadratic equation
• For exactly one solution, the quadratic must have a discriminant of zero
• This is the key insight that drives our solution strategy

3. Set up the intersection equation and SIMPLIFY

• Set the equations equal: \(\mathrm{4x + 1 = x^2 + px + 5}\)
• Rearrange to standard form: \(\mathrm{x^2 + (p - 4)x + 4 = 0}\)

4. INFER and apply the discriminant condition

• For exactly one solution: \(\mathrm{discriminant = 0}\)
• Using \(\mathrm{b^2 - 4ac = 0}\) where \(\mathrm{a = 1, b = (p - 4), c = 4}\)
• \(\mathrm{(p - 4)^2 - 4(1)(4) = 0}\)

5. SIMPLIFY the discriminant equation

• \(\mathrm{(p - 4)^2 - 16 = 0}\)
• \(\mathrm{(p - 4)^2 = 16}\)
• \(\mathrm{p - 4 = ±4}\)
• \(\mathrm{p = 8}\) or \(\mathrm{p = 0}\)

6. APPLY CONSTRAINTS to select the final answer

• Since p is positive: \(\mathrm{p = 8}\)

Answer: 8

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students don't connect "exactly one intersection point" to "discriminant equals zero." Instead, they might try to solve the system directly or guess-and-check with specific values of p. This leads to confusion and abandoning systematic solution, causing them to guess randomly.

Second Most Common Error:

Poor SIMPLIFY execution: Students correctly identify that discriminant = 0, but make algebraic errors when expanding \(\mathrm{(p - 4)^2}\) or solving the resulting equation. Common mistakes include getting the wrong sign when taking square roots or incorrectly expanding the squared term. This can lead to incorrect values of p and ultimately guessing.

The Bottom Line:

This problem requires recognizing that geometric conditions (one intersection point) translate to algebraic conditions (discriminant = 0). Students who miss this connection often get stuck trying less systematic approaches.