Hey whatup yall I need help and fast! Part B of the algebra semester b exam. If you could include all your work that would be great

Question

x/x-2 + x-1/x+1=-1

Answer 1

First, we need to get rid of the denominators by finding a common denominator. In this case, the common denominator is (x - 2)(x + 1)(x + 1)(x - 2), since it includes all of the denominators.

So, we begin by multiplying both sides of the equation by the common denominator:

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

Simplifying the left side, we get:

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

Multiplying out the terms, we get:

x(x^3 - 2x^2 + x - 1) + (x - 2)(x^3 + 2x^2 - x - 2) = -(x - 2)(x + 1)^2(x - 2)

Simplifying further, we get:

x^4 - x^3 - x^2 + 3x - 2 + x^4 - 3x^3 - 3x^2 + 5x + 4 = -x^4 + 4x^3 - 3x^2 - 6x + 4

Combining like terms, we get:

2x^4 - 7x^3 + 5x^2 + 2x + 2 = 0

Now, we need to solve for x. Unfortunately, this equation does not seem to have any easy solutions. We can try factoring, but this does not yield any obvious solutions:

(2x + 1)(x - 1)(x^2 - 5x - 2) = 0

So, we must resort to using the quadratic formula to solve for the last factor:

x = (5 ± sqrt(33))/2

Therefore, the solutions to the equation are:

x = -1/2, 1, (5 + sqrt(33))/2, (5 - sqrt(33))/2.