I know the answer of x/x-2 + x-1/x + 1; but I do not know how to explain it in words. The answer is x = 0,1, when I solve for x.

I really need this answer ASAP, if possible.

Nvm, I figured the answer out myself, thx for your help.

I assume the usual sloppiness with parentheses, and that your original equation was

x/(x-2) + (x-1)/x + 1 = 0

If you found x=0 as one solutions, you need to exclude it.

(x-1)/x is undefined when x=0

So, while it may appear that x=0 is a solution, it is not acceptable as a value in the original equation.

In addition, the roots of the above equation are 2/3 and 1, not 0.

If your equation was

x/(x-2) + (x-1)/(x+1) = 0

then there are no solutions at all.

So, I'm confused on just what you were after...

To explain the solution to the equation (x/x-2) + (x-1/x + 1) = 0, we can break it down step by step.

Step 1: Find the common denominator
To add fractions, we need a common denominator. In this case, the denominators are (x-2) and (x + 1). The common denominator will be (x-2)(x + 1).

Step 2: Multiply each fraction by the missing factor
To get the common denominator, we multiply the first fraction by (x + 1) and the second fraction by (x - 2).

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

Step 3: Combine the fractions
Now that we have a common denominator, we can combine the fractions.

[(x(x + 1)) + ((x - 1)(x - 2))]/((x - 2)(x + 1))
= (x^2 + x + x^2 - 3x + 2)/((x - 2)(x + 1))

Step 4: Simplify the expression
Next, simplify the numerator by combining like terms.

2x^2 - 2x + 2/((x - 2)(x + 1))

Step 5: Set the expression equal to zero and solve for x
We are given that the expression equals zero, so we can set it equal to zero and solve for x.

2x^2 - 2x + 2 = 0

Step 6: Solve the quadratic equation
To solve the quadratic equation, we can use factoring, the quadratic formula, or completing the square. However, the given equation does not factor, so the most straightforward approach is to use the quadratic formula:

x = (-b ± √(b^2 - 4ac)) / (2a)

In this case, a = 2, b = -2, and c = 2. Plugging the values into the formula:

x = (-(-2) ± √((-2)^2 - 4(2)(2)))/(2(2))
= (2 ± √(4 - 16))/4
= (2 ± √(-12))/4

We have a square root of a negative number, which means there are no real solutions to the equation. However, if we continue working with complex numbers, we can simplify the solution further.

x = (2 ± 2i√3)/4
= (1 ± i√3)/2

So, the solutions to the equation are x = (1 + i√3)/2 and x = (1 - i√3)/2.