# x/(x-2)- (x+1)/x= 8/(x^2-2x)

notice that x^2 - 2x factors to x(x-2), which are found as the denominators of the first two fractions.

Multiply each term by x(x-2), the rest is easy.

## To solve the equation x/(x-2) - (x+1)/x = 8/(x^2-2x), notice that the denominator x^2 - 2x can be factored as x(x-2). This is important because we can use this factorization to simplify the equation.

To begin, let's multiply each term in the equation by x(x-2). This step is taken to eliminate the denominators and make the equation easier to work with. The equation becomes:

x(x-2) * (x/(x-2)) - x(x-2) * ((x+1)/x) = x(x-2) * (8/(x^2-2x))

Now, simplify each term:

x(x) - x(x-2)(x+1) = 8(x)

Simplify further:

x^2 - x(x^2 - 3x + 2) = 8x

Distribute the terms:

x^2 - x^3 + 3x^2 - 2x - 8x = 0

Combine like terms:

-x^3 + 4x^2 - 10x = 0

Now, we have a cubic equation. To solve it, we can factor out an x:

x(-x^2 + 4x - 10) = 0

The equation will be true if either x = 0 or if the expression in the parentheses equals 0:

(-x^2 + 4x - 10) = 0

This is a quadratic equation. We can use factoring, completing the square, or the quadratic formula to solve for x. Once we find the solutions for x, we can substitute them back into the original equation to confirm their validity.