1993 AHSME Problem 20

Consider the equation 10z^2-3iz-k=0, where z is a complex variable and i^2=-1. Which of the following statements is true?

\begin{array}{l} \text{(A) For all positive real numbers k, both roots are pure imaginary} \quad\\ \text{(B) For all negative real numbers k, both roots are pure imaginary} \quad\\ \text{(C) For all pure imaginary numbers k, both roots are real and rational} \quad\\ \text{(D) For all pure imaginary numbers k, both roots are real and irrational} \quad\\ \text{(E) For all complex numbers k, neither root is real} \end{array}

Full credit goes to MAA for authoring these problems. These problems were taken on the AOPS website.