Subtract and Simplify:
4a^2/(a-b)-(4b^2-8ab)/(b-a)
I have the answer as 4(a+b)^2/(a-b)
or should it be 4(a-b)^2/(a-b)
just the numerator..
4a^2-8ab + 4b^2, right?
4(a-b)^2
Now does that reduce with the denominator?
Okay I have the answer as 4(a-b) is that right
What about
4a^2/(a-b)-(4b^2-8ab)/(b-a)
I thought that you would multiple the second part of the equation by -1
-(4b^2-8ab)/(b-a)
to get the common LCD of a-b
So I thought the equation would be 4a^2/(a-b) + (4b^2 + 8ab)/(a-b)
I have the answer as 4(a+b)^2/(a-b)
To subtract and simplify the given expression, let's start by simplifying each term individually:
1. Simplify the first term:
4a^2 / (a - b)
Since there are no like terms or common factors in the numerator and denominator, we cannot simplify this any further.
2. Simplify the second term:
(4b^2 - 8ab) / (b - a)
Factor out a common factor of 4 from the numerator:
4(b^2 - 2ab) / (b - a)
Now, let's factor out a negative sign from the denominator to match the first term:
4(b^2 - 2ab) / -(a - b)
Next, let's reverse the order of the terms inside the parentheses to match the first term:
4(b^2 - 2ab) / (b - a)
Now we have simplified both terms. Let's combine them:
4a^2 / (a - b) - 4(b^2 - 2ab) / (b - a)
Now, let's find the common denominator:
The common denominator is (a - b) * (b - a). Multiplying out the denominators gives (a - b) * (b - a) = (a^2 - ab - ab + b^2) = (a^2 - 2ab + b^2).
Now, rewrite each term with the common denominator:
[4a^2 * (b - a)] / [(a - b) * (b - a)] - [4(b^2 - 2ab) * (a - b)] / [(a - b) * (b - a)]
Now, simplify each term further:
[4ab - 4a^2] / (a^2 - 2ab + b^2) - [4ab^2 - 8a^2b - 4b^3 + 8ab^2] / (a^2 - 2ab + b^2)
Combine the numerators:
[4ab - 4a^2 - (4ab^2 - 8a^2b - 4b^3 + 8ab^2)] / (a^2 - 2ab + b^2)
Now, simplify the numerator:
4ab - 4a^2 - 4ab^2 + 8a^2b + 4b^3 - 8ab^2 / (a^2 - 2ab + b^2)
Combine like terms:
-4a^2 + 4ab + 8a^2b - 4ab^2 - 8ab^2 + 4b^3 / (a^2 - 2ab + b^2)
Now, we can simplify the numerator by combining like terms:
(4ab + 8a^2b - 4ab^2 - 4a^2 - 8ab^2 + 4b^3) / (a^2 - 2ab + b^2)
Now let's factor out the common factor (-4) from the numerator:
-4(a^2 + 2ab + ab^2 + a^2 + 2ab^2 - b^3) / (a^2 - 2ab + b^2)
Simplifying the expression inside the parentheses:
-4(2a^2 + 4ab + ab^2 - b^3) / (a^2 - 2ab + b^2)
Now, rearrange the terms inside the parentheses for easier factoring:
-4(a^2 - b^3 + 4ab + ab^2) / (a^2 -2ab + b^2)
Now we can see that both the numerator and the denominator have a common factor of (a^2 - 2ab + b^2). Let's cancel it out:
-4 / 1
Therefore, the simplified expression is -4.
So, the correct answer is not 4(a + b)^2 / (a - b) or 4(a - b)^2 / (a - b). It is simply -4.