Question:y = ax^2 + c\(\mathrm{y = 2(a + c)x}\)In the system of equations above, a and c are positive constants....

1. TRANSLATE the system into a single equation

• Given information:
  • \(\mathrm{y = ax^2 + c}\) (parabola)
  • \(\mathrm{y = 2(a + c)x}\) (line)
  • a and c are positive constants

• Since both expressions equal y, set them equal to each other:
  \(\mathrm{ax^2 + c = 2(a + c)x}\)

2. SIMPLIFY to standard quadratic form

• Move all terms to one side:
  \(\mathrm{ax^2 - 2(a + c)x + c = 0}\)

• This is now in standard form \(\mathrm{Ax^2 + Bx + C = 0}\) where:
  • \(\mathrm{A = a}\)
  • \(\mathrm{B = -2(a + c)}\)
  • \(\mathrm{C = c}\)

3. INFER what determines the number of solutions

• For any quadratic, the discriminant tells us about real solutions
• We need to calculate \(\mathrm{Δ = B^2 - 4AC}\)

4. SIMPLIFY the discriminant calculation

• \(\mathrm{Δ = [-2(a + c)]^2 - 4(a)(c)}\)

• \(\mathrm{Δ = 4(a + c)^2 - 4ac}\)

• \(\mathrm{Δ = 4[(a + c)^2 - ac]}\)

• \(\mathrm{Δ = 4[a^2 + 2ac + c^2 - ac]}\)

• \(\mathrm{Δ = 4[a^2 + ac + c^2]}\)

5. INFER the sign of the discriminant

• Since \(\mathrm{a \gt 0}\) and \(\mathrm{c \gt 0}\):
  • \(\mathrm{a^2 \gt 0}\) (positive)
  • \(\mathrm{c^2 \gt 0}\) (positive)
  • \(\mathrm{ac \gt 0}\) (positive)

• Therefore: \(\mathrm{a^2 + ac + c^2 \gt 0}\)
• This means: \(\mathrm{Δ = 4(a^2 + ac + c^2) \gt 0}\)

6. APPLY CONSTRAINTS using discriminant interpretation

• Since \(\mathrm{Δ \gt 0}\), the quadratic has two distinct real solutions

Answer: C. Two

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY skill: Making algebraic errors when expanding the discriminant

Students often get lost in the discriminant expansion, especially when dealing with:

• \(\mathrm{Δ = [-2(a + c)]^2 - 4ac}\)
• Forgetting that \(\mathrm{[-2(a + c)]^2 = 4(a + c)^2}\), not \(\mathrm{4(a + c)}\)
• Making sign errors or dropping terms during the expansion

This leads to an incorrect discriminant value and wrong conclusion about the number of solutions, causing them to select Choice A (Zero) or Choice B (One).

Second Most Common Error:

Missing conceptual knowledge: Not remembering that the discriminant determines the number of real solutions

Some students correctly set up the quadratic equation but then don't know what to do next. They might try to solve the quadratic directly or use other approaches instead of analyzing the discriminant. Without this key insight, they get stuck and resort to guessing among the answer choices.

The Bottom Line:

This problem tests whether students can systematically analyze a quadratic without actually solving it. The key insight is using the discriminant as a diagnostic tool, combined with careful algebraic manipulation and recognition that positive constants guarantee a positive discriminant.