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 like terms.
First, let's simplify the expression:
2/x^2 - 2x - x/2x - 4
Since the denominators, x^2 and 2x, are already different, we only need to combine the numerators:
2 - 2x - x
Now, let's combine like terms:
(2 - x) - 2x
Now, let's distribute the negative sign to the terms inside the parentheses:
2 - x - 2x
Next, let's combine like terms again:
2 - 3x
Therefore, the simplified expression is 2 - 3x.
To subtract and simplify the expression (2/x^2-2x) - (x/2x-4), we need to find a common denominator for the fractions.
The denominators are x^2-2x and 2x-4.
To find the common denominator, we need to factor both denominators.
Factor the first denominator:
x^2-2x = x(x-2)
Factor the second denominator:
2x-4 = 2(x-2)
Now, we can determine the common denominator, which is (x(x-2))(2(x-2)). Simplifying this expression, we get 2(x-2)(x(x-2)).
Now, let's rewrite the original expression with the common denominator:
(2/x^2-2x) - (x/2x-4) becomes (2(x-2)(x(x-2))/2(x-2)(x(x-2))) - (x(x-2)/2(x-2)(x(x-2)))
Simplifying further, we have:
= (2(x-2)(x(x-2)) - x(x-2)) / (2(x-2)(x(x-2)))
Expanding and combining like terms:
= (2x(x-2)^2 - x(x-2)) / (2(x-2)(x(x-2)))
Now, we can continue simplifying:
= (2x(x^2 - 4x + 4) - x^2 + 2x) / (2(x-2)(x(x-2)))
Expanding the multiplication:
= (2x^3 - 8x^2 + 8x - x^2 + 2x) / (2(x-2)(x(x-2)))
Combining like terms:
= (2x^3 - 9x^2 +10x) / (2(x-2)(x(x-2)))
Now, we can cancel out any common factors:
= x(2x^2 - 9x + 10) / (2(x-2)(x(x-2)))
The expression is now simplified.
To subtract and simplify the given expression (2/x^2-2x) - (x/2x-4), we need to follow these steps:
Step 1: Find a common denominator.
The denominators in this case are x^2-2x and 2x-4. To find a common denominator, we need to factorize both denominators.
x^2 - 2x can be factored as x(x - 2).
2x - 4 can be factored as 2(x - 2).
The common denominator is x(x - 2)(2).
Step 2: Rewrite the fractions with the common denominator.
For the first fraction, x^2-2x, we need to multiply the numerator and denominator by (2(x - 2)):
(x^2 - 2x) * (2(x - 2)) / (x(x - 2)(2))
For the second fraction, x, we need to multiply the numerator and denominator by (x):
(x * x) / (x(x - 2)(2))
After rewriting the fractions with the common denominator, our expression becomes:
(2(x^2 - 2x))/(x(x - 2)(2)) - (x^2)/(x(x - 2)(2))
Step 3: Simplify the expression.
For a subtraction problem like this, we need to combine the two fractions into one fraction. To do this, we can multiply the first fraction's numerator and denominator by -1 to make it negative.
(-(2(x^2 - 2x))/(x(x - 2)(2))) - (x^2)/(x(x - 2)(2))
Now we can combine the fractions by adding the numerators:
(-(2x^2 - 4x))/(x(x - 2)(2)) - (x^2)/(x(x - 2)(2))
Simplifying the numerator further:
-(2x^2 - 4x) - (x^2)/(x(x - 2)(2))
Distribute the negative sign:
-2x^2 + 4x - (x^2)/(x(x - 2)(2))
Combine like terms:
-2x^2 + 4x - (x^2)/(2x^2 - 4x)
Now we have a fully simplified expression.