Question
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
3x/x^2+2x-3 - 5x/x^2-4x+3
Answers
GPT 3.5
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.
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.