For the quadratic x^2 + x + 1 = 0, what is the nature of the roots?

For the quadratic x^2 + x + 1 = 0, what is the nature of the roots?

• A. Two distinct real roots
• B. No real roots (two distinct complex roots)
• C. One real root
• D. All real numbers

Answer: (B)

This question tests how the discriminant determines the nature of the roots. For x^2 + x + 1 = 0, Δ = b^2 − 4ac = 1 − 4 = −3, which is negative. A negative discriminant means there are no real roots; the solutions are complex and occur as a conjugate pair: x = (-1 ± i√3)/2. Those are two distinct complex numbers, so there are two complex roots and no real ones. The other possibilities would require Δ ≥ 0 or imply every real number solves the equation, which doesn’t hold here.

This question tests how the discriminant determines the nature of the roots. For x^2 + x + 1 = 0, Δ = b^2 − 4ac = 1 − 4 = −3, which is negative. A negative discriminant means there are no real roots; the solutions are complex and occur as a conjugate pair: x = (-1 ± i√3)/2. Those are two distinct complex numbers, so there are two complex roots and no real ones. The other possibilities would require Δ ≥ 0 or imply every real number solves the equation, which doesn’t hold here.