Solve.

x/x+3 - 3/x-3 = x^2+9/x^2-9

Answer: x = 0?

Plug your 0 into the x in your equation and you get 1 = -1, which is false, which mean: no, the answer is not x = 0.

Since you didn't show your work, I have no idea what you did wrong on your process... or maybe you just guess 0 and didn't do any work at all. But I'll tell you a few steps to approach this problem.

First, you find a common denominator for the 3 fractions (Hint: factor out the x^2-9). Now they all have the same denominator, you can basically ignore the denominator part and combine all the like-terms on the numerator and then bring "x" on one side and solve for x. Plug the number back into your equation to check and if it works, you know you got the right answer.

*I did it and I found the answer quite tricky, you sure you typed the problem correctly? Anyway, there's still an answer, just a tricky one.

Without parentheses your equation is ambiguous. I will assume that what is intended is

x/(x+3) - 3(x-3) = (x^2+9)/(x^2-9)
Multiply both sides by (x+3)(x-3), which is the same as x^2-9, and you get
x(x-3) -3(x+3) = x^2 +9
x^2 -6x -9 = x^2 + 9
-6x = 18
x = -3

x=0 is not a solution. You can verify that by substituing x=0 into the original equation. You get 1 = -1, which cannot be true

x= -3 result causes zeros to appear in denominators on both sides of the original equation, so technically speaking, the equation is meaningless there. Nevertheless, the equation tends toward validity as x approaches -3

I did the problem again and still got zero. I give up

To solve the given equation

(x / (x + 3)) - (3 / (x - 3)) = ((x^2 + 9) / (x^2 - 9))

we need to find the value(s) of x that satisfy the equation. Here's how we can solve it step by step:

Step 1: Remove the denominators by multiplying both sides of the equation by (x + 3)(x - 3)(x^2 - 9). This will clear the fractions.

(x(x - 3)(x^2 - 9) / (x + 3)) - (3(x + 3)(x^2 - 9) / (x - 3)) = x^2 + 9

Step 2: Simplify the expression on both sides:

(x(x - 3)(x^2 - 9) - 3(x + 3)(x^2 - 9)) / (x + 3) = x^2 + 9

Step 3: Distribute and expand the terms:

[x(x^3 - 3x^2 - 9x - 9) - 3(x^3 - 9x^2 - 27x - 81)] / (x + 3) = x^2 + 9

Step 4: Combine like terms and simplify further:

[x^4 - 3x^3 - 9x^2 - 9x - 27x^2 + 81x + 27x + 81] / (x + 3) = x^2 + 9

[x^4 - 3x^3 - 36x^2 + 72x + 81] / (x + 3) = x^2 + 9

Step 5: Multiply both sides of the equation by (x + 3) to get rid of the denominator:

x^4 - 3x^3 - 36x^2 + 72x + 81 = (x + 3)(x^2 + 9)

Step 6: Expand and simplify further:

x^4 - 3x^3 - 36x^2 + 72x + 81 = x^3 + 3x^2 + 9x + 27

Step 7: Rearrange the equation and combine like terms:

x^4 - 4x^3 - 39x^2 + 63x + 54 = 0

Step 8: Unfortunately, this equation is quite complex and does not have a simple factorization method. Solving it directly to find the value of x is not straightforward.

To continue solving the equation, we could use numerical methods such as graphing the equation and finding the x-intercepts, or using methods like Newton's method or the bisection method. However,-solving this equation manually can be quite time-consuming and challenging. In such cases, it's recommended to use computational tools like calculators or computer algebra systems to find the solutions.