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

1. TRANSLATE the intersection problem into mathematics

• Given information:
  • Line: \(\mathrm{y = 3x + 7}\)
  • Parabola: \(\mathrm{y = -2x^2 - 6x - 11}\)
  • Need: Number of intersection points

• To find intersections, set the y-values equal since both expressions equal y:
  \(\mathrm{3x + 7 = -2x^2 - 6x - 11}\)

2. SIMPLIFY to get standard quadratic form

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

• Multiply by -1 for easier work:
  \(\mathrm{2x^2 + 9x + 18 = 0}\)

3. INFER the solution strategy

• This is now a quadratic equation in standard form \(\mathrm{ax^2 + bx + c = 0}\)
• We don't need to find exact x-values - just whether real solutions exist
• The discriminant will tell us this: \(\mathrm{\Delta = b^2 - 4ac}\)

4. SIMPLIFY the discriminant calculation

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

5. INFER the final answer from the discriminant

• Since \(\mathrm{\Delta = -63 \lt 0}\), there are no real solutions to the quadratic
• No real solutions means no intersection points between the graphs

Answer: (A) 0

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY execution: Students make arithmetic errors when rearranging terms or computing the discriminant.

For example, they might incorrectly combine -6x - 3x as -3x instead of -9x, or miscalculate \(\mathrm{4(2)(18)}\) as something other than 144. These errors lead to a different discriminant value that could be positive, suggesting intersection points exist.

This may lead them to select Choice (B) (1) or Choice (C) (2).

Second Most Common Error:

Missing conceptual knowledge: Students don't remember or understand what the discriminant tells us about the number of solutions.

They might correctly set up and simplify the equation but then try to solve \(\mathrm{2x^2 + 9x + 18 = 0}\) directly using factoring or the quadratic formula, not realizing the discriminant shortcut. This leads to unnecessary complex calculations and potential confusion.

This leads to confusion and guessing.

The Bottom Line:

This problem tests whether students can connect the geometric concept of graph intersections to the algebraic concept of simultaneous equation solutions, then use discriminant analysis as an efficient solution method.