-x/(x-3) + 2x/(x+2) (x-1) x (x-2) (x+2)/(x-3)(x-3)
my question is should i multiply first and then do the adding or subtracting through a LCD or should i do the adding first?
If you use some parentheses to properly specify the numerators and denominators, I think you have a typo and meant
-x/(x-3) + 2x/((x+2)(x-1)) * ((x-2)(x+2))/((x-3)(x+3))
As usual, do multiplication before addition. In effect you have
-x + y * z
Do the y*z first, giving you
-x/(x-3) + (2x(x-2)(x+2))/((x+2)(x-1)(x-3)(x+3))
= -x/(x-3) + (2x(x-2))/((x-1)(x-3)(x+3))
Nor you need to get the GCD before you can do the addition. That gives you
(-x(x-1)(x+3) + 2x(x-2))/((x-1)(x-3)(x+3))
(-x^3-x)/((x-1)(x-3)(x+3))
= -(x(x^2+1))/((x-1)(x-3)(x+3))
To simplify the expression, you first need to find a common denominator (LCD) for each fraction in the expression. In this case, the LCD is (x - 3)(x - 3)(x - 2)(x + 2).
To determine whether to multiply or add/subtract first, it's usually more straightforward to first simplify each fraction separately, and then combine them. Let's break down the steps:
Step 1: Simplify each fraction separately.
Rewrite the original expression:
-x/(x - 3) + 2x/(x + 2) * (x - 1) * x * (x - 2) * (x + 2) / (x - 3)(x - 3)
Now, simplify each fraction:
For the first fraction, -x/(x - 3), there's not much further simplification possible.
For the second fraction, 2x/(x + 2), you can multiply through by the (x - 1) term since denominator already includes (x + 2).
So, 2x/(x + 2) * (x - 1) becomes 2x(x - 1)/(x + 2).
Step 2: Combine the fractions.
Substitute the simplified fractions back into the original expression:
-x/(x - 3) + 2x(x - 1)/(x + 2) * (x - 1) * x * (x - 2) * (x + 2) / (x - 3)(x - 3)
Now you have two fractions, -x/(x - 3) and 2x(x - 1)/(x + 2), with a common denominator of (x - 3)(x - 3)(x - 2)(x + 2). You can now add them together.
Step 3: Add the fractions.
Since the denominators are the same, you can combine the numerators:
(-x + 2x(x - 1)(x - 1)(x - 2)(x + 2)) / (x - 3)(x - 3)(x - 2)(x + 2)
Simplifying further, distribute 2x into (x - 1)(x - 1)(x - 2)(x + 2):
(-x + 2(x^2 - 2x + 1)(x - 2)(x + 2)) / (x - 3)(x - 3)(x - 2)(x + 2)
Simplify within the parentheses:
(-x + 2(x^4 - 4x^3 + 4x^2 - 4x - 4)(x + 2)) / (x - 3)(x - 3)(x - 2)(x + 2)
Expand and combine like terms within the numerator:
(-x + 2x^4 - 8x^3 + 8x^2 - 8x - 8)(x + 2) / (x - 3)(x - 3)(x - 2)(x + 2)
Continuing to simplify:
(2x^4 - 8x^3 + 8x^2 - 9x - 8)(x + 2) / (x - 3)(x - 3)(x - 2)(x + 2)
At this point, you'll notice that the denominator (x + 2) cancels out with the numerator, resulting in:
2x^4 - 8x^3 + 8x^2 - 9x - 8 / (x - 3)(x - 3)(x - 2)
So, after simplifying, you end up with the expression:
(2x^4 - 8x^3 + 8x^2 - 9x - 8) / (x - 3)(x - 3)(x - 2)
To summarize, the steps to simplify the original expression:
1. Simplify each fraction separately.
2. Combine the fractions by adding/subtracting.
3. Simplify further if possible.
Keep in mind that these steps can vary depending on the expression. It's always important to analyze the given expression and determine the most efficient approach.