How do you simplify this?
(x+3/x-6)-2/(x+3/x-6)+8
simplify fractions first:
(x+3)/(x-6) - 2(x-6)/(x+3) + 8
now put all over common denominator of (x-6)(x+3)
(x+3)(x+3) - 2(x-6)(x-6) + 8(x+3)(x-6)
---------------------------------------
(x-6)(x+3)
= (7x^2+6x-207)//(x^2-3x-18)
The above answer is not really a satisfying "simplification." How about this?
x+3/x-6 = 1 + 9/(x-6)
x-6/x+3 = 1 - 9/(x+3)
so, we have
1 + 9/(x-6) + 2(1 - 9/(x+3)) + 8
9/(x-6) - 18/(x+3) + 11
"-" got lost.
Of course, I meant
9/(x-6) + 18/(x+3) + 7
To simplify the given expression, we need to follow the order of operations (PEMDAS/BODMAS) and combine like terms, if possible.
Step 1: Simplify the first fraction
The first fraction in the expression is (x + 3) / (x - 6). Since there are no like terms to combine, we leave this fraction as it is.
(x + 3) / (x - 6) - 2 / (x + 3) + 8
Step 2: Simplify the second fraction
The second fraction in the expression is 2 / (x + 3). Again, there are no like terms, so we keep this fraction as is.
(x + 3) / (x - 6) - 2 / (x + 3) + 8
Step 3: Apply the distributive property to the third term
The third term is 8, and we need to distribute it to both fractions.
8 * (x + 3) / (x - 6) = (8x + 24) / (x - 6)
So, the expression becomes:
(x + 3) / (x - 6) - 2 / (x + 3) + (8x + 24) / (x - 6)
Step 4: Find a common denominator
To combine the fractions, we need to find a common denominator. The common denominator is (x - 6)(x + 3) since it contains both denominators.
(x + 3) / (x - 6) - 2 / (x + 3) + (8x + 24) / (x - 6)
= [(x + 3)(x + 3)] / [(x - 6)(x + 3)] - [2(x - 6)] / [(x + 3)(x - 6)] + (8x + 24) / (x - 6)
Step 5: Combine the fractions
Now, we can combine the fractions over the common denominator.
[(x + 3)(x + 3)] / [(x - 6)(x + 3)] - [2(x - 6)] / [(x + 3)(x - 6)] + (8x + 24) / (x - 6)
= [(x + 3)(x + 3) - 2(x - 6) + (8x + 24)] / [(x - 6)(x + 3)]
Step 6: Simplify the numerator
Expand and simplify the numerator.
[(x + 3)(x + 3) - 2(x - 6) + (8x + 24)] / [(x - 6)(x + 3)]
= [x^2 + 6x + 9 - 2x + 12 + 8x + 24] / [(x - 6)(x + 3)]
= [x^2 + 12x + 45] / [(x - 6)(x + 3)]
Therefore, the simplified form of the given expression is (x^2 + 12x + 45) / ((x - 6)(x + 3)).