Basic Algebra/Rational Expressions and Equations/Adding and Subtracting When the Denominators are Different
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If you have to add two rational fractions with different denominators, as the first step, you have to find the LCM:
3 + 3 x+1 x-1
LCM = (x+1)(x-1)
Now divide the LCM by both denominators and multiply by their respectives numerators:
(x+1)(x-1) / (x+1) = (x-1) . (3) = 3x-3 (x+1)(x-1) / (x-1) = (x+1) . (3) = 3x+3
The sum of the two results would be the new nominator:
3x-3+3x+3= (x+1)(x-1)
6x (x+1)(x-1)
This is another example:
6x + 9x 2x-6 x2-6x+9
We factorize both denominators and find the LCM
2x-6 = 2(x-3) x2-6x+9 = (x-3)2 LCM = 2(x-3)2
Now we divide and multiply:
2(x-3)2 / 2(x-3) = 2x2-12x+18 / 2x-6 = x-3 (x-3) . 6x = 6x2-18x
2(x-3)2 / (x-3)2 = 2x2-12x+18 / x2-6x+9 = 2 (2) . (9x) = 18x
We add the results to obtain the nominator; the denominator is the LCM:
6x2-18x+18x= 2(x-3)2
6x2 2(x-3)2
We can factorize the nominator to simplify the result:
2(3x2) =2(x-3)2
3x2 (x-3)2
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