How many distinct real solutions does the given equation have?x^2 - 12x + 27 = 0

1. TRANSLATE the problem information

• Given equation: \(\mathrm{x^2 - 12x + 27 = 0}\)
• Need to find: How many distinct real solutions exist
• This is asking about the nature of solutions, not the actual values

2. INFER the most efficient approach

• Rather than solve completely, we can use the discriminant to determine solution count
• For any quadratic \(\mathrm{ax^2 + bx + c = 0}\), the discriminant \(\mathrm{\Delta = b^2 - 4ac}\) tells us:
  • If \(\mathrm{\Delta \gt 0}\): exactly two distinct real solutions
  • If \(\mathrm{\Delta = 0}\): exactly one real solution
  • If \(\mathrm{\Delta \lt 0}\): zero real solutions

3. Identify coefficients and SIMPLIFY the discriminant calculation

• From \(\mathrm{x^2 - 12x + 27 = 0}\): \(\mathrm{a = 1, b = -12, c = 27}\)
• Calculate: \(\mathrm{\Delta = b^2 - 4ac = (-12)^2 - 4(1)(27)}\)
• SIMPLIFY: \(\mathrm{\Delta = 144 - 108 = 36}\)

4. INFER the final answer from the discriminant value

• Since \(\mathrm{\Delta = 36 \gt 0}\), the equation has exactly two distinct real solutions

Answer: A. Exactly two

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY execution: Students make arithmetic errors when calculating the discriminant, especially with the negative coefficient \(\mathrm{b = -12}\).

Common mistakes include:

• Calculating \(\mathrm{(-12)^2}\) as -144 instead of +144
• Computing \(\mathrm{4(1)(27)}\) incorrectly as something other than 108
• Making sign errors in the final subtraction

These calculation errors can lead to an incorrect discriminant value, causing them to select Choice B (Exactly one) if they get \(\mathrm{\Delta = 0}\), or Choice C (Zero) if they incorrectly get a negative result.

Second Most Common Error:

Missing conceptual knowledge: Students don't remember the discriminant rules or confuse the conditions.

They might remember that the discriminant is important but mix up which discriminant values correspond to which solution counts. This conceptual confusion leads to randomly selecting among the first three choices rather than systematically determining the answer.

The Bottom Line:

This problem tests both computational accuracy and conceptual understanding of the discriminant. Success requires careful arithmetic with negative numbers and solid recall of discriminant interpretation rules.