(x+3)/(x^2-2x-8)- (x-5)/(x^2-12x+32)
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What is "The villages?" What does math have to do with studying villages?
I will take a giant leap of faith and assume you want it simplified,
(x+3)/(x^2-2x-8)- (x-5)/(x^2-12x+32)
= (x+3)/((x-4)(x+2)) - (x-5)/((x-4)(x-8))
LCD is (x-4)(x+2)(x-8)
= [(x+3)(x-8) - (x-5)(x+2)]/((x-4)(x+2)(x-8) )
= ...
expand the top and simplify / (x-4)(x+2)(x-8)
To simplify the expression (x+3)/(x^2-2x-8) - (x-5)/(x^2-12x+32), we need to find a common denominator for both fractions and then combine the numerators.
Step 1: Factorize the denominators
The first denominator, x^2-2x-8, can be factored as (x-4)(x+2).
The second denominator, x^2-12x+32, can be factored as (x-8)(x-4).
Step 2: Write the expression with the common denominator
The common denominator of (x-4)(x+2)(x-8) can be used for both fractions. So, the simplified expression becomes:
[(x+3)(x-8)]/[(x-4)(x+2)(x-8)] - [(x-5)(x+2)]/[(x-4)(x+2)(x-8)]
Step 3: Combine the numerators
To combine the numerators, we need to distribute and simplify each term.
[(x+3)(x-8)] - [(x-5)(x+2)]
= (x^2 - 5x + 24) - (x^2 - 3x - 10)
= x^2 - 5x + 24 - x^2 + 3x + 10
= -2x + 34
Step 4: Simplify the expression
We now have the simplified numerator (-2x + 34) over the common denominator (x-4)(x+2)(x-8). Therefore, the simplified expression is:
(-2x + 34)/[(x-4)(x+2)(x-8)]
And there you have it, the simplified expression of (x+3)/(x^2-2x-8) - (x-5)/(x^2-12x+32) is (-2x + 34)/[(x-4)(x+2)(x-8)]