Simplify the difference, type your answer in factored form and use a comma to separate if needed. State any restrictions on the variable.
3x/x^2+2x-3 - 5x/x^2-4x+3
To simplify the difference, we find a common denominator and then subtract the fractions.
The denominator of the first fraction is x^2 + 2x - 3, which can be factored as (x + 3)(x - 1).
The denominator of the second fraction is x^2 - 4x + 3, which can be factored as (x - 3)(x - 1).
So the common denominator is (x + 3)(x - 1)(x - 3).
Now we can rewrite the fractions with the common denominator:
3x/(x + 3)(x - 1) - 5x/(x - 3)(x - 1)
Multiplying the numerator and denominator of the first fraction by (x - 3) and multiplying the numerator and denominator of the second fraction by (x + 3), we get:
(3x(x - 3))/(x + 3)(x - 1)(x - 3) - (5x(x + 3))/(x - 3)(x - 1)(x + 3)
Simplifying the numerators, we have:
(3x^2 - 9x)/(x + 3)(x - 1)(x - 3) - (5x^2 + 15x)/(x - 3)(x - 1)(x + 3)
Now that the fractions have the same denominator, we can subtract them:
(3x^2 - 9x - 5x^2 - 15x)/(x + 3)(x - 1)(x - 3)
Simplifying the numerator, we get:
(-2x^2 - 24x)/(x + 3)(x - 1)(x - 3)
Factoring out -2x from the numerator, we have:
-2x(x + 12)/(x + 3)(x - 1)(x - 3)
So the simplified difference is -2x(x + 12)/(x + 3)(x - 1)(x - 3).
Restrictions on the variable:
The denominator cannot be zero, so x cannot be -3, 1, or 3.
To simplify the difference (subtraction):
Step 1: Factor the denominators:
x^2 + 2x - 3 = (x + 3)(x - 1)
x^2 - 4x + 3 = (x - 1)(x - 3)
Step 2: Rewrite the expression with the factored denominators:
3x/(x + 3)(x - 1) - 5x/(x - 1)(x - 3)
Step 3: Find the common denominator, which is (x + 3)(x - 1)(x - 3).
Step 4: Rewrite the expression with the common denominator:
(3x(x - 3) - 5x(x + 3))/((x + 3)(x - 1)(x - 3))
Step 5: Simplify the numerator:
(3x^2 - 9x - 5x^2 - 15x)/((x + 3)(x - 1)(x - 3))
Step 6: Combine like terms in the numerator:
(-2x^2 - 24x)/((x + 3)(x - 1)(x - 3))
Step 7: Factor out a 2x from the numerator:
-2x(x + 12)/((x + 3)(x - 1)(x - 3))
So, the simplified difference in factored form is -2x(x + 12)/((x + 3)(x - 1)(x - 3)).
Restrictions on the variable:
The given expression is defined for all real values of x except x = -3, x = 1, and x = 3, as these values would make the denominator(s) equal to zero, resulting in an undefined expression.
To simplify the given expression, we will start by factoring the denominators and finding the LCD (Least Common Denominator).
For the first fraction, x^2 + 2x - 3, we need to find two numbers whose sum is 2 (coefficient of the x-term) and whose product is -3 (constant term). The numbers that satisfy these conditions are 3 and -1. Thus, we can factor the denominator as (x + 3)(x - 1).
For the second fraction, x^2 - 4x + 3, we need to find two numbers whose sum is -4 (coefficient of the x-term) and whose product is 3 (constant term). The numbers that satisfy these conditions are -3 and -1. Thus, we can factor the denominator as (x - 3)(x - 1).
Now, we have the following expression:
3x/(x + 3)(x - 1) - 5x/(x - 3)(x - 1)
To combine the fractions, we need to find the common denominator, which is (x + 3)(x - 1)(x - 3). Multiplying the numerator and denominator of each fraction by the necessary factors to obtain the common denominator, we get:
(3x(x - 3) - 5x(x + 3))/[(x + 3)(x - 1)(x - 3)]
Expanding and simplifying the numerator, we have:
(3x^2 - 9x - 5x^2 - 15x)/[(x + 3)(x - 1)(x - 3)]
Combining like terms, we get:
(-2x^2 - 24x)/[(x + 3)(x - 1)(x - 3)]
The final simplified expression, in factored form, is:
(-2x(x + 12))/[(x + 3)(x - 1)(x - 3)]
And the restrictions on the variable are x ≠ -3, 1, and 3, since these values would make the denominator zero, resulting in undefined expressions.