Answer:
Bellow
Explanation:
To prove the equation:
(x - 1)/(x - 3) - (x ^ 2 - x + 2)/(x ^ 2 - 2x - 3) = 1/(x + 1)
we need to find a common denominator for the left-hand side of the equation. The common denominator is (x - 3)(x + 1)(x - 3), which is the product of all the factors in the denominators.
Using this common denominator, we can rewrite the left-hand side of the equation as:
[(x - 1)(x + 1)(x - 3) - (x ^ 2 - x + 2)(x + 1)] / [(x - 3)(x + 1)(x - 3)]
Simplifying the numerator using distributive property, we get:
[(x^3 - 2x^2 - 2x + 2) - (x^3 - 2x^2 - x - 2)] / [(x - 3)(x + 1)(x - 3)]
Simplifying the numerator further, we get:
(-x + 4) / [(x - 3)(x + 1)(x - 3)]
Now, we can simplify the right-hand side of the equation by multiplying both the numerator and the denominator by (x - 3)(x + 1), which gives us:
1 / (x + 1) = (x - 3)(x + 3) / [(x - 3)(x + 1)(x - 3)]
Therefore, the original equation can be written as:
(-x + 4) / [(x - 3)(x + 1)(x - 3)] = (x - 3)(x + 3) / [(x - 3)(x + 1)(x - 3)]
Cancelling the common factors on both sides, we get:
-x + 4 = x^2 - 9
Rearranging and simplifying, we get:
x^2 + x - 13 = 0
This is a quadratic equation that can be solved using the quadratic formula:
x = (-1 ± √(1 + 4*13)) / 2
Simplifying, we get:
x = (-1 ± √53) / 2
Therefore, we have proved that the equation:
(x - 1)/(x - 3) - (x ^ 2 - x + 2)/(x ^ 2 - 2x - 3) = 1/(x + 1)
holds true for all values of x except (-1, 3, 3 - √13, 3 + √13).