For what value of k does the equation x2 + kx + 9have:

Two distinct real roots
One real root
No real root

One real root.

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 ?

6 > k > -6 no roots?

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

To determine the value of k for each case, we need to consider the discriminant of the quadratic equation. The discriminant is given by the expression b^2 - 4ac, where a, b, and c are the coefficients of the quadratic equation ax^2 + bx + c = 0.

1. Two distinct real roots:
For a quadratic equation to have two distinct real roots, the discriminant should be greater than zero. In this case, the equation is x^2 + kx + 9. So, we have:
Discriminant = b^2 - 4ac = k^2 - 4(1)(9) > 0
Simplifying the inequality gives k^2 - 36 > 0.
To solve this inequality, we can factor it: (k + 6)(k - 6) > 0.
To find when this inequality is true, we consider the signs of the factors:
- When (k + 6) > 0 and (k - 6) > 0, we have k > 6.
- When (k + 6) < 0 and (k - 6) < 0, we have k < -6.
Therefore, the values of k that make the equation have two distinct real roots are k > 6 and k < -6.

2. One real root:
For a quadratic equation to have one real root, the discriminant should be equal to zero. So, we have:
Discriminant = b^2 - 4ac = k^2 - 4(1)(9) = 0.
Simplifying the equation gives k^2 - 36 = 0.
We can solve this equation by factoring: (k + 6)(k - 6) = 0.
Setting each factor equal to zero gives us two possible values:
- k + 6 = 0, which implies k = -6.
- k - 6 = 0, which implies k = 6.
Therefore, when k equals -6 or 6, the equation will have one real root.

3. No real root:
For a quadratic equation to have no real root, the discriminant should be less than zero. In this case, the equation is x^2 + kx + 9. So, we have:
Discriminant = b^2 - 4ac = k^2 - 4(1)(9) < 0
Simplifying the inequality gives k^2 - 36 < 0.
To solve this inequality, we can factor it: (k + 6)(k - 6) < 0.
To find when this inequality is true, we consider the signs of the factors:
- When (k + 6) > 0 and (k - 6) < 0, we have -6 < k < 6.
- When (k + 6) < 0 and (k - 6) > 0, we have k < -6 or k > 6.
Therefore, the values of k that make the equation have no real roots are -6 < k < 6.

To summarize:
- For two distinct real roots, k > 6 or k < -6.
- For one real root, k = -6 or k = 6.
- For no real roots, -6 < k < 6.