How do you solve #15/x-15/(x-2)= -2#?

To solve the equation 15/x - 15/(x-2) = -2, you can follow these steps:

1. Find a common denominator for the fractions on the left side of the equation. The common denominator is x(x-2).

2. Multiply each term by the common denominator to eliminate the fractions. This gives you: 15(x-2) - 15x = -2x(x-2).

3. Simplify both sides of the equation by distributing and combining like terms. This results in: 15x - 30 - 15x = -2x^2 + 4x.

4. Combine like terms on both sides of the equation. The x terms cancel out, leaving you with: -30 = -2x^2 + 4x.

5. Rearrange the equation to bring all terms to one side, setting it equal to zero: -2x^2 + 4x + 30 = 0.

6. To solve this quadratic equation, you can either factor it or use the quadratic formula. In this case, factoring is not possible, so we'll use the quadratic formula: x = (-b ± √(b^2 - 4ac)) / (2a).

7. Substitute the values from the equation into the quadratic formula: x = (-4 ± √(4^2 - 4(-2)(30))) / (2(-2)).

8. Simplify the equation inside the square root: x = (-4 ± √(16 + 240)) / (-4).

9. Further simplify: x = (-4 ± √256) / (-4).

10. Take the square root of 256: x = (-4 ± 16) / (-4).

11. Simplify the equation: x = (12 or 4).

Therefore, the solutions to the equation 15/x - 15/(x-2) = -2 are x = 12 and x = 4.