3x^2 - 15x + 18 = 0 How many distinct real solutions are there to the given equation?...

1. TRANSLATE the problem information

• Given equation: \(3\mathrm{x}^2 - 15\mathrm{x} + 18 = 0\)
• Need to find: How many distinct real solutions exist
• This is a quadratic equation in standard form \(\mathrm{ax}^2 + \mathrm{bx} + \mathrm{c} = 0\)

2. INFER the approach

• For any quadratic equation, the number of real solutions depends on the discriminant
• The discriminant \(\mathrm{b}^2 - 4\mathrm{ac}\) tells us:
  • If positive: exactly two distinct real solutions
  • If zero: exactly one real solution
  • If negative: zero real solutions
• We need to calculate the discriminant for this specific equation

3. Identify the coefficients

• From \(3\mathrm{x}^2 - 15\mathrm{x} + 18 = 0\):
  • \(\mathrm{a} = 3\)
  • \(\mathrm{b} = -15\)
  • \(\mathrm{c} = 18\)

4. SIMPLIFY to calculate the discriminant

• Discriminant = \(\mathrm{b}^2 - 4\mathrm{ac}\)
• Substitute values: \((-15)^2 - 4(3)(18)\)
• Calculate: \(225 - 216 = 9\)

5. INFER the final answer

• Since discriminant = \(9 \gt 0\), the equation has exactly two distinct real solutions

Answer: B. Exactly two

Why Students Usually Falter on This Problem

Most Common Error Path:

Conceptual gap: Not remembering the discriminant formula or its relationship to the number of solutions

Students might try to solve the quadratic equation directly (factoring, quadratic formula) rather than using the discriminant. This takes much longer and isn't necessary to answer the question. They may get confused during the solving process and select the wrong answer or run out of time.

This leads to confusion and guessing among the answer choices.

Second Most Common Error:

Weak SIMPLIFY skill: Making arithmetic errors when calculating the discriminant

Students correctly identify that they need the discriminant but make calculation mistakes:

• Forgetting that \(\mathrm{b} = -15\), so \(\mathrm{b}^2 = (-15)^2 = 225\) (not -225)
• Miscalculating \(4(3)(18) = 216\)
• Making sign errors in the final subtraction

This may lead them to get a negative discriminant and select Choice D (Zero) or a zero discriminant and select Choice A (Exactly one).

The Bottom Line:

This problem tests whether students know the discriminant shortcut for determining the number of real solutions without actually solving the quadratic. Many students default to solving completely, which is unnecessary and error-prone for this question type.