Question
Simplify the difference and type the answer in factored form. State any restrictions on the variable.
2x/x^2-2x-15 - 6x/x^2-8x+15
2x/x^2-2x-15 - 6x/x^2-8x+15
Answers
GPT 3.5
To simplify the difference, we first need to factor the denominators of each rational expression.
For the first rational expression, we have x^2 - 2x - 15. This can be factored as (x - 5)(x + 3).
For the second rational expression, we have x^2 - 8x + 15. This can be factored as (x - 3)(x - 5).
Now, we can rewrite the original expression:
2x/(x - 5)(x + 3) - 6x/(x - 3)(x - 5)
Next, we need to find a common denominator. The two denominators have common factors of (x - 5), so we only need to consider the other factors.
The least common denominator (LCD) is (x - 3)(x - 5)(x + 3).
To rewrite the expression with the common denominator, we need to multiply the numerator and denominator of each rational expression by the missing factors from the LCD.
For the first rational expression, we need to multiply the numerator and denominator by (x - 3):
(2x * (x - 3))/((x - 5)(x + 3)(x - 3)) = (2x^2 - 6x)/((x - 5)(x + 3)(x - 3))
Similarly, for the second rational expression, we need to multiply the numerator and denominator by (x + 3):
(6x * (x + 3))/((x - 3)(x - 5)(x + 3)) = (6x^2 + 18x)/((x - 3)(x - 5)(x + 3))
Now, we can subtract the two rational expressions:
(2x^2 - 6x)/((x - 5)(x + 3)(x - 3)) - (6x^2 + 18x)/((x - 3)(x - 5)(x + 3))
To combine the two terms, we need a common denominator. Since the denominators are already the same, we can write the difference as a single rational expression:
(2x^2 - 6x - (6x^2 + 18x))/((x - 3)(x - 5)(x + 3))
Now, let's simplify the numerator:
2x^2 - 6x - 6x^2 - 18x = -4x^2 - 24x
Thus, the simplified difference is:
(-4x^2 - 24x)/((x - 3)(x - 5)(x + 3))
Restrictions on the variable: The restrictions on the variable occur when the denominator becomes zero. In this case, the restrictions are x = 3, x = 5, and x = -3.
For the first rational expression, we have x^2 - 2x - 15. This can be factored as (x - 5)(x + 3).
For the second rational expression, we have x^2 - 8x + 15. This can be factored as (x - 3)(x - 5).
Now, we can rewrite the original expression:
2x/(x - 5)(x + 3) - 6x/(x - 3)(x - 5)
Next, we need to find a common denominator. The two denominators have common factors of (x - 5), so we only need to consider the other factors.
The least common denominator (LCD) is (x - 3)(x - 5)(x + 3).
To rewrite the expression with the common denominator, we need to multiply the numerator and denominator of each rational expression by the missing factors from the LCD.
For the first rational expression, we need to multiply the numerator and denominator by (x - 3):
(2x * (x - 3))/((x - 5)(x + 3)(x - 3)) = (2x^2 - 6x)/((x - 5)(x + 3)(x - 3))
Similarly, for the second rational expression, we need to multiply the numerator and denominator by (x + 3):
(6x * (x + 3))/((x - 3)(x - 5)(x + 3)) = (6x^2 + 18x)/((x - 3)(x - 5)(x + 3))
Now, we can subtract the two rational expressions:
(2x^2 - 6x)/((x - 5)(x + 3)(x - 3)) - (6x^2 + 18x)/((x - 3)(x - 5)(x + 3))
To combine the two terms, we need a common denominator. Since the denominators are already the same, we can write the difference as a single rational expression:
(2x^2 - 6x - (6x^2 + 18x))/((x - 3)(x - 5)(x + 3))
Now, let's simplify the numerator:
2x^2 - 6x - 6x^2 - 18x = -4x^2 - 24x
Thus, the simplified difference is:
(-4x^2 - 24x)/((x - 3)(x - 5)(x + 3))
Restrictions on the variable: The restrictions on the variable occur when the denominator becomes zero. In this case, the restrictions are x = 3, x = 5, and x = -3.