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For the equation 10z² - 3iz - k = 0 in the complex variable z (with i² = -1), which statement is correct?

1. For every real positive k, both roots are purely imaginary.
2. For every negative real k, both roots are purely imaginary.
3. For every purely imaginary k, both roots are real and irrational.
4. For every complex k, neither root is real.

Correct answer: For every negative real k, both roots are purely imaginary.

Solution

Putting z = iy turns the equation into a real quadratic 10y² - 3y + k = 0; for negative real k its discriminant 9 - 40k > 0, so real y exist and both roots z = iy are purely imaginary.

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