In the equation -{x^2 + 4x - 3 = k} where k is a constant, the equation has no real...

1. TRANSLATE the problem information

• Given equation: \(-\mathrm{x}^2 + 4\mathrm{x} - 3 = \mathrm{k}\)
• Need to find: Values of k where equation has no real solutions
• What "no real solutions" means: The discriminant must be negative

2. INFER the solution approach

• To analyze real solutions, I need the discriminant
• First, I must rearrange to standard form \(\mathrm{ax}^2 + \mathrm{bx} + \mathrm{c} = 0\)
• Then apply discriminant condition: \(\Delta \lt 0\)

3. SIMPLIFY to get standard form

• Start with: \(-\mathrm{x}^2 + 4\mathrm{x} - 3 = \mathrm{k}\)
• Move k to left side: \(-\mathrm{x}^2 + 4\mathrm{x} - 3 - \mathrm{k} = 0\)
• Multiply by -1: \(\mathrm{x}^2 - 4\mathrm{x} + (3 + \mathrm{k}) = 0\)
• Now in standard form with \(\mathrm{a} = 1, \mathrm{b} = -4, \mathrm{c} = 3 + \mathrm{k}\)

4. SIMPLIFY the discriminant calculation

• Discriminant formula: \(\Delta = \mathrm{b}^2 - 4\mathrm{ac}\)
• Substitute values: \(\Delta = (-4)^2 - 4(1)(3 + \mathrm{k})\)
• Calculate: \(\Delta = 16 - 4(3 + \mathrm{k})\)
  \(\Delta = 16 - 12 - 4\mathrm{k}\)
  \(\Delta = 4 - 4\mathrm{k}\)

5. APPLY CONSTRAINTS for no real solutions

• For no real solutions: \(\Delta \lt 0\)
• Set up inequality: \(4 - 4\mathrm{k} \lt 0\)
• Solve: \(4 \lt 4\mathrm{k}\), so \(\mathrm{k} \gt 1\)

6. INFER the correct answer choice

• Check each option against \(\mathrm{k} \gt 1\):
• (A) -1: Not greater than 1
• (B) 0: Not greater than 1
• (C) 1: Equal to 1, not greater
• (D) 2: Greater than 1 ✓

Answer: D

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students don't recognize that "no real solutions" connects to discriminant analysis. Instead, they might try to solve the equation directly or substitute answer choices randomly.

This leads to confusion and guessing because they lack a systematic approach to determine when quadratic equations have no solutions.

Second Most Common Error:

Poor SIMPLIFY execution: Students make algebraic errors when rearranging to standard form or calculating the discriminant, particularly with the signs when multiplying by -1 or distributing the -4.

For example, they might get the discriminant as \(4 + 4\mathrm{k}\) instead of \(4 - 4\mathrm{k}\), leading to \(\mathrm{k} \lt -1\) instead of \(\mathrm{k} \gt 1\). This would make them incorrectly select Choice (A) (-1).

The Bottom Line:

This problem tests whether students understand the connection between discriminant signs and solution existence, plus their ability to handle algebraic manipulation with negative coefficients. The key insight is recognizing that "no real solutions" translates to a discriminant condition, not a direct solving approach.