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Solve the following equation. Show all your work.
xx−2+x−1x+1=−1
(2 points)
To solve the equation, we will first simplify the left side of the equation:
x(x-2) + (x-1)/(x+1) = -1
Expanding the first term:
x^2 - 2x + (x-1)/(x+1) = -1
Now, let's combine like terms:
x^2 - 2x + (x-1)/(x+1) + 1 = 0
Next, let's find a common denominator for the fraction:
x^2 - 2x + (x(x+1) - 1)/(x+1) + 1 = 0
Simplify the fraction:
x^2 - 2x + (x^2 + x - 1)/(x+1) + 1 = 0
Combine the fractions:
x^2 - 2x + (2x^2 + x - 1)/(x+1) + 1 = 0
Now, simplify the equation:
3x^2 - x/(x+1) = -1
Multiply both sides by (x+1) to get rid of the fraction:
3x^2(x+1) - x = -x - 1
3x^2 + 3x - x + 1 = 0
3x^2 + 2x + 1 = 0
Now, we have a quadratic equation. Let's solve for x using the quadratic formula:
x = [-2 ± sqrt((2)^2 - 4(3)(1))] / 2(3)
x = [-2 ± sqrt(4 - 12)] / 6
x = [-2 ± sqrt(-8)] / 6
x = [-2 ± 2i√2] / 6
x = (-1 ± i√2) / 3
Therefore, the solutions to the equation are:
x1 = (-1 + i√2) / 3
x2 = (-1 - i√2) / 3