For any real value of k, the roots of the quadratic equation x² + kx - 9 = 0 are :

1. real and equal

2. imaginary

3. real and unequal

4. none of these

D = b² - 4ac

Substituting values we get :

D = k² - 4 × 1 x (-9)

= k² + 36

For any real value of k, k² + 36 is a positive number.

∴ D > 0

∴ Roots are real and unequal.

Hence, Option 3 is the correct option.