Solve the following equation for the variable x.
(4/x-2)+(3/x+2)=(-3x/x^2-4)
Thank you very much for your help.
To solve the equation (4 / (x - 2)) + (3 / (x + 2)) = (-3x / (x^2 - 4)) for the variable x, we will:
1. Simplify the equation and get rid of the denominators.
2. Rearrange the equation to isolate the variable.
3. Solve for x.
Let's go through each step in detail:
Step 1: Simplify the equation and get rid of the denominators.
To eliminate the denominators, we need to find the least common multiple (LCM) of the denominators (x - 2), (x + 2), and (x^2 - 4). The LCM of these three expressions is (x^2 - 4). Therefore, we can multiply every term in the equation by (x^2 - 4).
(x^2 - 4) * [(4 / (x - 2)) + (3 / (x + 2))] = (x^2 - 4) * (-3x / (x^2 - 4))
Expanding each term on both sides:
4(x^2 - 4) / (x - 2) + 3(x^2 - 4) / (x + 2) = -3x
Now, let's simplify further:
4(x^2 - 4) = 3(x - 2)(x + 2) * [x^2 - 4 is canceled out]
4x^2 - 16 = 3(x^2 - 4)
4x^2 - 16 = 3x^2 - 12
Now, rearranging the equation to isolate the variable:
4x^2 - 3x^2 = -12 + 16
x^2 = 4
Step 2: Solve for x.
To solve for x, we take the square root of both sides of the equation:
√(x^2) = √4
Simplifying further:
x = ±2
Therefore, the values of x that satisfy the equation are x = 2 and x = -2.
So, the solution to the given equation is x = 2 and x = -2.