Simplify the difference. State every possible restictions of the variable. Show your work.
4x/x^2+4x-5 - 6x/x^2-6x+5
To simplify the given expression:
4x/(x^2 + 4x - 5) - 6x/(x^2 - 6x + 5)
First, we factor the denominators to determine their possible restrictions:
x^2 + 4x - 5 = (x - 1)(x + 5)
x^2 - 6x + 5 = (x - 1)(x - 5)
So the denominators cannot equal zero. Therefore, the possible restrictions are:
x ≠ 1 and x ≠ -5 for the first fraction
x ≠ 1 and x ≠ 5 for the second fraction
Next, we find a common denominator for the two fractions, which is (x - 1)(x + 5)(x - 5):
4x(x - 1)(x + 5)/(x^2 + 4x - 5)(x - 1)(x + 5) - 6x(x - 1)(x + 5)/(x^2 - 6x + 5)(x - 1)(x + 5)
Simplifying further by canceling out common factors in the numerator and denominator:
[4x(x - 1)(x + 5) - 6x(x - 1)(x + 5)] / [(x^2 + 4x - 5)(x^2 - 6x + 5)(x - 1)(x + 5)]
Combining like terms in the numerator:
[4x^3 + 4x^2 + 20x - 4x^2 + 4x - 20x - 6x^3 + 6x^2 + 30x] / [(x^2 + 4x - 5)(x^2 - 6x + 5)(x - 1)(x + 5)]
Simplifying further:
[-2x^3 + 16x^2 + 54x] / [(x^2 + 4x - 5)(x^2 - 6x + 5)(x - 1)(x + 5)]
Therefore, the simplified expression is (-2x^3 + 16x^2 + 54x) / (x^2 + 4x - 5)(x^2 - 6x + 5)(x - 1)(x + 5).
The possible restrictions on the variable x, as mentioned earlier, are x ≠ 1, x ≠ -5, and x ≠ 5.
To simplify the difference (4x/x^2+4x-5) - (6x/x^2-6x+5), we need to find a common denominator for both fractions.
The denominators are x^2+4x-5 and x^2-6x+5. To find the common denominator, we can factorize the denominators:
x^2 + 4x - 5 = (x + 5)(x - 1)
x^2 - 6x + 5 = (x - 5)(x - 1)
The common denominator is (x + 5)(x - 1)(x - 5).
Now, let's rewrite the fractions with the common denominator:
4x(x - 5) / [(x + 5)(x - 1)(x - 5)] - 6x(x + 5) / [(x + 5)(x - 1)(x - 5)]
Next, we can combine the numerators:
[4x(x - 5) - 6x(x + 5)] / [(x + 5)(x - 1)(x - 5)]
Simplifying the numerator:
[4x^2 - 20x - 6x^2 - 30x] / [(x + 5)(x - 1)(x - 5)]
Combining like terms in the numerator:
[-2x^2 - 50x] / [(x + 5)(x - 1)(x - 5)]
Finally, we can simplify further:
-2x(x + 25) / [(x + 5)(x - 1)(x - 5)]
Now, let's consider the possible restrictions of the variable x. Looking at the factors in the denominator, we can see that x cannot equal -5, 1, or 5, as it would make the denominator equal to zero. Therefore, the restrictions on x are x ≠ -5, x ≠ 1, and x ≠ 5.
To simplify the difference (also known as subtraction), we need to find a common denominator for the two fractions. Let's start by factoring the denominators:
x^2 + 4x - 5 can be factored as (x + 5)(x - 1)
x^2 - 6x + 5 can be factored as (x - 1)(x - 5)
Now, we can rewrite the expression with the factored denominators:
(4x)/(x + 5)(x - 1) - (6x)/(x - 1)(x - 5)
To find a common denominator, we multiply the denominators together:
Common denominator: (x + 5)(x - 1)(x - 5)
Now, we can rewrite the fractions with the common denominator:
(4x) * (x - 5) / [(x + 5)(x - 1)(x - 5)] - (6x) * (x + 5) / [(x - 1)(x - 5)(x + 5)]
Next, we simplify each fraction:
(4x)(x - 5) - (6x)(x + 5) / [(x + 5)(x - 1)(x - 5)]
Expanding and simplifying:
4x^2 - 20x - 6x^2 - 30x / [(x + 5)(x - 1)(x - 5)]
Combining like terms:
(-2x^2 - 50x) / [(x + 5)(x - 1)(x - 5)]
Finally, we can see that the simplified difference is (-2x^2 - 50x) / [(x + 5)(x - 1)(x - 5)].
Now, let's discuss the restrictions on the variable x. These restrictions arise when the denominator becomes zero, as division by zero is undefined.
From the factored denominators, we can see that x cannot equal -5, 1, or 5, as substituting these values would make the denominator zero. Therefore, the restrictions on the variable x are x ≠ -5, x ≠ 1, and x ≠ 5.
Overall, the simplified difference is (-2x^2 - 50x) / [(x + 5)(x - 1)(x - 5)], with restrictions on the variable x: x ≠ -5, x ≠ 1, and x ≠ 5.