Consider the line y = 3x + 7 and the parabola y = -2x^2 - 6x - 11. How many...

1. TRANSLATE the intersection problem into mathematical language

• Given information:
  • Line: \(\mathrm{y = 3x + 7}\)
  • Parabola: \(\mathrm{y = -2x^2 - 6x - 11}\)
  • Need to find: Number of intersection points
• At intersection points, both equations have the same y-value for the same x-value

2. INFER the solution approach

• Set the equations equal to each other: \(\mathrm{3x + 7 = -2x^2 - 6x - 11}\)
• This will give us a quadratic equation whose solutions are the x-coordinates of intersection points
• The number of real solutions tells us the number of intersection points

3. SIMPLIFY to standard form

• Start with: \(\mathrm{3x + 7 = -2x^2 - 6x - 11}\)
• Move all terms to one side: \(\mathrm{2x^2 + 6x + 3x + 7 + 11 = 0}\)
• Combine like terms: \(\mathrm{2x^2 + 9x + 18 = 0}\)

4. INFER that the discriminant determines the number of solutions

• For quadratic \(\mathrm{ax^2 + bx + c = 0}\), the discriminant is \(\mathrm{Δ = b^2 - 4ac}\)
• If \(\mathrm{Δ < 0}\): no real solutions (no intersection points)
• If \(\mathrm{Δ = 0}\): one real solution (one intersection point)
• If \(\mathrm{Δ > 0}\): two real solutions (two intersection points)

5. SIMPLIFY the discriminant calculation

• With \(\mathrm{a = 2, b = 9, c = 18}\):
• \(\mathrm{Δ = (9)^2 - 4(2)(18)}\)
  \(\mathrm{= 81 - 144}\)
  \(\mathrm{= -63}\)

Since \(\mathrm{Δ = -63 < 0}\), there are no real solutions.

Answer: (A) 0

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students might try to solve each equation separately rather than setting them equal to find intersection points.

Instead of recognizing that intersections occur where y-values are equal, they might try to find where each equation equals zero or attempt some other approach. This leads to confusion about what the problem is actually asking and typically results in guessing.

Second Most Common Error:

Poor SIMPLIFY execution: Students make algebraic mistakes when rearranging to standard form or calculating the discriminant.

Common errors include sign mistakes when moving terms, combining like terms incorrectly (getting \(\mathrm{2x^2 + 8x + 18 = 0}\) instead of \(\mathrm{2x^2 + 9x + 18 = 0}\)), or arithmetic errors in the discriminant calculation. These errors often lead to a positive discriminant, causing them to select Choice (C) 2.

The Bottom Line:

This problem tests whether students can connect the geometric concept of intersection with the algebraic process of solving equations. The key insight is that intersection points correspond to solutions of a quadratic equation, and the discriminant reveals how many solutions exist without actually solving.