Which quadratic equation has no real solutions?

1. TRANSLATE the problem requirements

• We need to find which quadratic equation has no real solutions
• Each equation is in standard form \(\mathrm{ax^2 + bx + c = 0}\)
• We need to examine all four choices systematically

2. INFER the mathematical approach

• The discriminant \(\mathrm{b^2 - 4ac}\) determines the nature of solutions:
  • Positive discriminant → two distinct real solutions
  • Zero discriminant → exactly one real solution
  • Negative discriminant → no real solutions
• We need the equation with negative discriminant

3. TRANSLATE coefficients for each choice

• Choice A: \(\mathrm{x^2 + 14x - 49 = 0}\) → \(\mathrm{a = 1, b = 14, c = -49}\)
• Choice B: \(\mathrm{x^2 - 14x + 49 = 0}\) → \(\mathrm{a = 1, b = -14, c = 49}\)
• Choice C: \(\mathrm{5x^2 - 14x - 49 = 0}\) → \(\mathrm{a = 5, b = -14, c = -49}\)
• Choice D: \(\mathrm{5x^2 - 14x + 49 = 0}\) → \(\mathrm{a = 5, b = -14, c = 49}\)

4. SIMPLIFY discriminant calculations

Choice A: \(\mathrm{b^2 - 4ac = 14^2 - 4(1)(-49)}\)
\(\mathrm{= 196 + 196}\)
\(\mathrm{= 392 \gt 0}\)

Choice B: \(\mathrm{b^2 - 4ac = (-14)^2 - 4(1)(49)}\)
\(\mathrm{= 196 - 196}\)
\(\mathrm{= 0}\)

Choice C: \(\mathrm{b^2 - 4ac = (-14)^2 - 4(5)(-49)}\)
\(\mathrm{= 196 + 980}\)
\(\mathrm{= 1176 \gt 0}\)

Choice D: \(\mathrm{b^2 - 4ac = (-14)^2 - 4(5)(49)}\)
\(\mathrm{= 196 - 980}\)
\(\mathrm{= -784 \lt 0}\)

5. INFER the final answer

• Only Choice D produces a negative discriminant (-784)
• Therefore, only Choice D has no real solutions

Answer: D

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY execution: Students make sign errors when calculating the discriminant, especially with negative coefficients. The most common mistake is incorrectly handling \(\mathrm{(-14)^2}\) or the negative signs in \(\mathrm{-4ac}\) terms.

For example, they might calculate Choice D as: \(\mathrm{(-14)^2 - 4(5)(49)}\)
\(\mathrm{= -196 - 980}\)
\(\mathrm{= -1176}\), forgetting that \(\mathrm{(-14)^2 = +196}\), not -196. Or they might write \(\mathrm{4(5)(49)}\) as -980 instead of +980. These arithmetic errors can lead them to incorrect discriminant signs and wrong conclusions about which equations have real solutions.

This leads to confusion and potentially selecting any of the wrong answer choices.

The Bottom Line:

This problem requires systematic application of the discriminant test to all four choices, with careful attention to arithmetic involving negative numbers. Success depends on both knowing the discriminant rules and executing the calculations accurately.