Question

Show your work and answer
x over x-2 + x-1 over x+1 = -1

Answer 1

To solve this equation, we need to find a common denominator for the two fractions on the left side. The common denominator in this case is (x-2)(x+1).

So, let's rewrite the equation with the common denominator:
x(x+1)/(x-2)(x+1) + (x-1)(x-2)/(x-2)(x+1) = -1

Now, we combine the numerators:
[x(x+1) + (x-1)(x-2)] / (x-2)(x+1) = -1

Now, we simplify the numerator:
[x^2 + x + x^2 - 3x + 2] / (x-2)(x+1) = -1
[2x^2 - 2x + 2] / (x-2)(x+1) = -1

Multiplying both sides by (x-2)(x+1) to get rid of the denominator:
2x^2 - 2x + 2 = -1(x^2 - x)

Now, expand and simplify both sides:
2x^2 - 2x + 2 = -x^2 + x
3x^2 - 3x + 2 = 0

Now, we have a quadratic equation. We can solve this by using the quadratic formula:
x = [-(-3) ± √((-3)^2 - 4(3)(2))] / 2(3)
x = [3 ± √(9 - 24)] / 6
x = [3 ± √(-15)] / 6
x = (3 ± i√15) / 6

Therefore, the solutions to the equation x over x-2 + x-1 over x+1 = -1 are:
x = (3 + i√15) / 6 or x = (3 - i√15) / 6.