Simplify the difference. State every possible restictions of the variable. Show your work.
2x/x^2-2x-15 - 6x/x^2-8x+15
To simplify the difference, let's find a common denominator for both fractions.
The denominators of the fractions are (x^2 - 2x - 15) and (x^2 - 8x + 15). Since the denominators are both quadratic expressions, we can factor them to simplify.
The first denominator, x^2 - 2x - 15, can be factored as (x - 5)(x + 3).
The second denominator, x^2 - 8x + 15, can be factored as (x - 3)(x - 5).
Now, the simplified expression becomes:
2x/(x - 5)(x + 3) - 6x/(x - 3)(x - 5)
To find a common denominator, we multiply the numerator and denominator of the first fraction by (x - 3) and the numerator and denominator of the second fraction by (x + 3):
(2x(x - 3))/[(x - 5)(x + 3)(x - 3)] - (6x(x + 3))/[(x - 3)(x - 5)(x + 3)]
Simplifying the numerators gives:
(2x^2 - 6x)/[(x - 5)(x + 3)(x - 3)] - (6x^2 + 18x)/[(x - 3)(x - 5)(x + 3)]
Now that we have a common denominator, we can subtract the fractions:
(2x^2 - 6x - 6x^2 - 18x)/[(x - 5)(x + 3)(x - 3)]
Combining like terms in the numerator gives:
(-4x^2 - 24x)/[(x - 5)(x + 3)(x - 3)]
Finally, we can simplify by factoring out -4x from the numerator:
-4x(x + 6)/[(x - 5)(x + 3)(x - 3)]
So, the simplified difference is -4x(x + 6)/[(x - 5)(x + 3)(x - 3)].
Now let's determine the possible restrictions on the variable. The expression will be undefined if any of the denominators equal zero.
Restrictions:
1) x - 5 ≠ 0 (x ≠ 5)
2) x + 3 ≠ 0 (x ≠ -3)
3) x - 3 ≠ 0 (x ≠ 3)
Therefore, the variable x cannot be equal to 5, -3, or 3.
To simplify the given expression, start by factoring the denominators:
x^2 - 2x - 15:
This expression can be factored as (x - 5)(x + 3).
x^2 - 8x + 15:
This expression can be factored as (x - 3)(x - 5).
Now, let's rewrite the expression with the factored denominators:
2x/(x - 5)(x + 3) - 6x/(x - 3)(x - 5)
To combine the fractions, we need to find the least common denominator (LCD). The LCD is (x - 5)(x + 3)(x - 3). Now, let's rewrite both fractions with the LCD:
2x(x - 3)/(x - 5)(x + 3)(x - 3) - 6x(x + 3)/(x - 3)(x - 5)(x - 3)
Simplifying further:
(2x^2 - 6x)/(x - 5)(x + 3)(x - 3)
Now, let's look at the restrictions of the variable:
Since the denominators contain (x - 5), (x + 3), and (x - 3), the given expression would be undefined when any of these factors equal zero. Therefore, the restrictions on the variable are x ≠ 5, x ≠ -3, and x ≠ 3.
The simplified expression is (2x^2 - 6x)/(x - 5)(x + 3)(x - 3), with restrictions x ≠ 5, x ≠ -3, and x ≠ 3.
To simplify the given expression and determine any possible restrictions on the variables, we will first factor the denominators and look for any common factors.
Given expression:
(2x / (x^2 - 2x - 15)) - (6x / (x^2 - 8x + 15))
Step 1: Factor the denominators.
The denominator of the first fraction can be factored as:
x^2 - 2x - 15 = (x - 5)(x + 3)
The denominator of the second fraction can be factored as:
x^2 - 8x + 15 = (x - 5)(x - 3)
The common factor is (x - 5), so we can simplify the expression as follows:
(2x / (x - 5)(x + 3)) - (6x / (x - 5)(x - 3))
Step 2: Combine the fractions.
To combine the fractions, we need a common denominator. In this case, the common denominator is formed by multiplying the two denominators together:
Common denominator = (x - 5)(x + 3)(x - 3)
Now we can rewrite the expression with the common denominator:
[(2x * (x - 3)) - (6x * (x + 3))] / [(x - 5)(x + 3)(x - 3)]
Simplifying the numerator, we get:
(2x^2 - 6x - 6x^2 - 18x) / [(x - 5)(x + 3)(x - 3)]
Step 3: Combine like terms in the numerator.
(2x^2 - 6x - 6x^2 - 18x) becomes (-4x^2 - 24x)
The expression is now:
(-4x^2 - 24x) / [(x - 5)(x + 3)(x - 3)]
Step 4: Look for any possible restrictions on the variable.
In order to find any restrictions on the variable, we need to examine the denominator.
The expression will be undefined if any of the factors in the denominator become zero.
(x - 5)(x + 3)(x - 3) = 0
Setting each factor equal to zero and solving for x, we have:
x - 5 = 0 --> x = 5
x + 3 = 0 --> x = -3
x - 3 = 0 --> x = 3
Therefore, the possible restrictions on the variable are x = 5, x = -3, and x = 3.
The simplified expression is:
(-4x^2 - 24x) / [(x - 5)(x + 3)(x - 3)]
Restrictions on the variable: x = 5, x = -3, x = 3