x/x-2 - x+1/x = 8/x^2-2x
notice the last denominator factors to x(x-2) and those factors are found in the other two fractions
so x(x-2) is the lowest common denominator.
now multiply each term by that x(x-2) to get
(x)(x) -(x+1)(x-2) = 8
simplify and solve
I got x=6
To solve the given equation, start by finding the lowest common denominator for all the fractions involved. In this case, the lowest common denominator is x(x-2) because it contains both factors present in the denominators.
Next, you need to multiply each term in the equation by the lowest common denominator, which is x(x-2), to clear the fractions. This is done to ensure that all the denominators are the same. Multiplying each term by x(x-2) gives:
x(x)/(x-2) - (x+1)(x-2)/x = 8/(x(x-2))
Now, simplify the equation:
(x^2)/(x-2) - (x^2 - x - 2)/x = 8/(x(x-2))
To simplify further, multiply the terms within the parentheses:
(x^2)/(x-2) - (x^2 - x - 2)/x = 8/(x(x-2))
(x^2)/(x-2) - (x^2 - x - 2)/x = 8/(x^2 - 2x)
Now, you can combine like terms:
[(x^2)(x) - (x^2 - x - 2)(x-2)]/[(x-2)(x)] = 8/(x^2 - 2x)
Simplify the numerator:
[x^3 - (x^3 - x^2 - 2x - 2x^2)])/[(x-2)(x)] = 8/(x^2 - 2x)
Simplify further:
[x^3 - x^3 + x^2 + 2x + 2x^2]/[(x-2)(x)] = 8/(x^2 - 2x)
Combine like terms:
(3x^2 + 2x)/[(x-2)(x)] = 8/(x^2 - 2x)
Now, cross multiply to eliminate the fractions:
(3x^2 + 2x)(x^2 - 2x) = 8(x-2)(x)
Expand the equation:
3x^4 - 6x^3 + 2x^3 - 4x^2 + 2x^2 - 4x - 6x + 12 = 8x^2 - 16x
Simplify:
3x^4 - 4x^3 - 2x^2 - 10x + 12 = 8x^2 - 16x
Now, combine like terms and move all terms to one side of the equation:
3x^4 - 4x^3 - 2x^2 - 8x^2 + 6x + 16x + 12 = 0
Simplify further:
3x^4 - 4x^3 - 10x^2 + 22x + 12 = 0
Finally, solve the equation. Since this is a fourth-degree polynomial, it may not have a simple algebraic solution. You may need to use a numerical method or a graphing calculator to find the solution. In your case, you obtained x = 6 as the solution.