Question

For what value of k does the equation x2 + kx + 9have:
Two distinct real roots
One real root
No real root

Answer 1

One real root.

Answer 2

First of all you don't have an equation, you must have meant:
x^2 + kx + 9 = 0

For two distinct real roots, the discriminant b^2 - 4ac > 0
k^2 - 4(1)(9) > 0
k^2 > 36
±k > 6
k > 6 or k < -6

To have 1 root, the discriminant b^2 - 4ac must be zero
k^2 - 4(1)(9) = 0
k^2 = 36
k = ± 6

So what do you think must be the value of b^2 - 4ac to have no real roots ?

Answer 3

6 > k > -6 no roots?

Answer 4

yes, I would prefer you last statement to say

-6 < k < 6

I know it is the same thing as your answer, but traditionally let's go
from left to right . Visualize the number line, negatives on the left.
for -10 < x < 10 , I visualize x to be between -10 and + 10