x cube plus kx plus 9=0

I took it as: x^3+kx+9=0
I answered:
If so.........

x^3+kx+9=0

Subtract 9 from both sides (Somenoe correct me if I'm wrong, please) which looks like:

x^3+kx+9-9=0-9
Now the best part.... finding variables

x^3+kx=9

x^3+(k•x)=9
Hmmm...

x^3= (x•x•x)

x•x•x•x+k=9

I'm not sure how to proceed....
How would I do this? I want to learn while I'm answering questions...
Thank you!
- Anonymous

To solve the equation x^3 + kx + 9 = 0, you can use algebraic techniques such as factoring or using the rational root theorem. Let's explore the steps further:

Step 1: Subtract 9 from both sides of the equation to isolate the terms involving x:

x^3 + kx = -9

Step 2: Next, we want to express x^3 + kx as a product of factors. One way to do this is by factoring out the greatest common factor of x:

x(x^2 + k) = -9

Step 3: Now, we have two cases to consider:

Case 1: If x = 0, then the equation becomes:

0(x^2 + k) = -9
0 = -9

This is not a valid solution, so we move to the second case.

Case 2: If x ≠ 0, then we can divide both sides of the equation by x:

x^2 + k = -9/x

Step 4: At this point, the equation has become a quadratic equation in terms of x^2. To solve for x^2, we can use various methods like factoring, completing the square, or using the quadratic formula. Let's use the quadratic formula:

x^2 = (-9/x) - k

Applying the quadratic formula:

x^2 = (-9/x) - k

x = [-(9/x) ± sqrt((9/x)^2 - 4k)] / 2

Simplifying the expression under the square root:

x = [-(9/x) ± sqrt((81/x^2) - 4k)] / 2

Step 5: Finally, we have the potential solutions for x. To determine if they are valid solutions, substitute them back into the original equation and check if it satisfies the equation x^3 + kx + 9 = 0.

Remember, this is just one method to approach this equation. Depending on the values of k and other factors, different methods might be more suitable. It's always helpful to have a good understanding of algebraic techniques and practice with solving various types of equations.