X over X+3 subtract X over X-3 = X^2 + 9 over X^2 -9
solve for x. show all work thank you
To solve the equation (X/X+3) - (X/X-3) = (X^2 + 9)/(X^2 -9) for x, we need to find a common denominator and simplify the expression on both sides of the equation.
Let's go through the steps:
Step 1: Determine the common denominator:
The denominators are (X+3) and (X-3). The common denominator will be the product of these two denominators: (X+3)(X-3).
Step 2: Rewrite the equation with the common denominator:
[(X)(X-3)/(X+3)(X-3)] - [(X)(X+3)/(X+3)(X-3)] = (X^2 + 9)/(X^2 -9)
Step 3: Simplify the numerators:
[X(X-3) - X(X+3)] / (X+3)(X-3) = (X^2 + 9)/(X^2 -9)
Simplifying the numerators, we get:
[X^2 - 3X - (X^2 + 3X)] / (X+3)(X-3) = (X^2 + 9)/(X^2 -9)
Combining like terms in the numerator:
[X^2 - 3X - X^2 - 3X] / (X+3)(X-3) = (X^2 + 9)/(X^2 -9)
Simplifying further, we have:
[-6X] / (X+3)(X-3) = (X^2 + 9)/(X^2 -9)
Step 4: Cross-multiply:
Multiply both sides of the equation by (X+3)(X-3) to eliminate the denominator:
[-6X] = (X^2 + 9)(X+3)(X-3)/(X^2 -9)
Step 5: Simplify the right side:
Using the identity (a^2 - b^2) = (a + b)(a - b), we can simplify the right side further:
[-6X] = [(X^2 + 9)(X+3)(X-3)] / [(X+3)(X-3)]
Canceling out the common factors:
[-6X] = (X^2 + 9)
Step 6: Move all terms to one side:
Bring all the terms to the left side of the equation:
[-6X] - (X^2 + 9) = 0
Step 7: Simplify and combine like terms:
Combine the terms:
-X^2 - 6X - 9 = 0
Step 8: Solve the quadratic equation:
To solve this equation, we can use factoring, completing the square, or the quadratic formula. In this case, let's use factoring.
Factor the quadratic equation:
-(X^2 + 6X + 9) = 0
-(X + 3)^2 = 0
Now, set each factor equal to zero:
X + 3 = 0
Solving for X, we get:
X = -3
Therefore, the solution to the given equation is X = -3.