Subtract and simplify, make sure your answer is either an integer or a fraction. Show your work.
2/x^2-2x - x/2x-4
To subtract and simplify the expression 2/x^2-2x - x/2x-4, we need to find a common denominator and combine the fractions.
The common denominator is (x^2-2x)(2x-4).
For the first fraction, we need to multiply the numerator and denominator by (2x-4) to obtain the common denominator:
2(2x-4)/(x^2-2x)(2x-4) = (4x-8)/(x^2-2x)(2x-4)
For the second fraction, we need to multiply the numerator and denominator by (x^2-2x) to obtain the common denominator:
x(x^2-2x)/(2x-4)(x^2-2x) = x(x^2-2x)/(x^2-2x)(2x-4)
Now, we can subtract the fractions:
(4x-8)/(x^2-2x)(2x-4) - x(x^2-2x)/(x^2-2x)(2x-4)
Combine the numerators:
(4x-8 - x(x^2-2x))/(x^2-2x)(2x-4)
Simplify the numerator:
(4x-8 - x^3 + 2x^2)/(x^2-2x)(2x-4)
Combine like terms:
(-x^3 + 2x^2 + 4x - 8)/(x^2-2x)(2x-4)
Now, the expression is simplified.
To subtract and simplify the expression (2/x^2 - 2x) - (x/2x - 4), we can first simplify each fraction separately.
For the first fraction, 2/x^2 - 2x, there is no simplification possible.
For the second fraction, x/2x - 4, we can simplify by dividing both the numerator and denominator by x to get (1/2 - 4/x).
Now, let's rewrite the expression with the simplified fractions:
(2/x^2 - 2x) - (x/2x - 4) = 2/x^2 - 2x - (1/2 - 4/x)
To simplify further, let's find a common denominator. The common denominator of x^2 and 2 is 2x^2.
So, multiply the first fraction 2/x^2 by 2/2 to get (4/2x^2), and multiply the second fraction (1/2 - 4/x) by x/x to get [(x - 4x^2) / (2x^2)].
Now we have the expression:
(4/2x^2) - (x - 4x^2) / (2x^2)
To subtract the fractions, we need a common denominator, which is 2x^2.
Thus, we have:
(4 - (x - 4x^2)) / (2x^2)
Simplifying further, we get:
(4 - x + 4x^2) / (2x^2)
Now the expression is simplified.
To subtract and simplify the given expression, we need to combine the two fractions into one by finding a common denominator.
The denominators of the two fractions are x^2 - 2x and 2x - 4. To find a common denominator, we need to factorize both denominators.
Let's start with x^2 - 2x:
x^2 - 2x = x(x - 2)
Now, let's factorize 2x - 4:
2x - 4 = 2(x - 2)
From the factorization, we see that both denominators have a common factor of (x - 2). So, the common denominator is (x - 2) multiplied by any remaining factors from each denominator.
Our common denominator is (x - 2) * x * 2 = 2x(x - 2).
Now, let's rewrite the fractions with the common denominator:
2/x^2 - 2x = (2 * 2)/[2x(x - 2)]
x/2x - 4 = (x * x)/[2x(x - 2)]
Next, we can subtract the two fractions:
[(2 * 2) - (x * x)]/[2x(x - 2)]
= (4 - x^2)/[2x(x - 2)]
Finally, we can simplify the expression if possible. In this case, the numerator (4 - x^2) is a difference of squares, so we can factorize it:
(4 - x^2) = (2 + x)(2 - x)
Substituting this back into our expression:
[(2 + x)(2 - x)]/[2x(x - 2)]
This is our final simplified answer.