Problem:
Consider the equation , where is a complex variable and . Which of the following statements is true?
Answer Choices:
A. For all positive real numbers , both roots are pure imaginary.
B. For all negative real numbers , both roots are pure imaginary.
C. For all pure imaginary numbers , both roots are real and rational.
D. For all pure imaginary numbers , both roots are real and irrational.
E. For all complex numbers , neither root is real.
Solution:
Use the quadratic formula to obtain , which has discriminant . If , then , so is false. If is a negative real number, then is a negative real number, so is true. If , then , and the roots are and , so and are false. If (which is a complex number), then the roots are and , so is false.