If b^2 - 4ac < 0, what is the nature of the roots?

If b^2 - 4ac < 0, what is the nature of the roots?

• A. b^2 - 4ac < 0
• B. a^2 + b^2 ≠ c^2
• C. b^2 - 4ac > 0
• D. b^2 - 4ac = 0

Answer: (A)

The key idea is how the discriminant controls the roots of a quadratic. For a quadratic with real coefficients, the discriminant is Δ = b^2 − 4ac. When Δ < 0, the roots are non-real—specifically a pair of complex conjugates. Since you’re given Δ < 0, the nature is that the roots are non-real. Among the options, the one that states the same condition, b^2 − 4ac < 0, directly matches this situation. The other statements describe different possibilities: Δ > 0 would give two real roots, Δ = 0 would give a repeated real root, and a^2 + b^2 ≠ c^2 is unrelated to the discriminant.

The key idea is how the discriminant controls the roots of a quadratic. For a quadratic with real coefficients, the discriminant is Δ = b^2 − 4ac. When Δ < 0, the roots are non-real—specifically a pair of complex conjugates.

Since you’re given Δ < 0, the nature is that the roots are non-real. Among the options, the one that states the same condition, b^2 − 4ac < 0, directly matches this situation. The other statements describe different possibilities: Δ > 0 would give two real roots, Δ = 0 would give a repeated real root, and a^2 + b^2 ≠ c^2 is unrelated to the discriminant.