Simplify the following
[2x / (3x (2nd power) - 5x - 2] + [(x-1) / (x (2nd power) -x - 20)]
let's write it this way
2x/(3x^2 - 5x - 2) + (x-1)/(x^2 -x - 20)
= 2x/[(3x+1)(x-2)] + (x-1)/[(x-5)(x+4)]
Your LCD is (3x+1)(x-2)(x-5)(x+4)
Unusual, as I had expected one of the factors to be the same in the denominators.
anyway...
[2x(x-5)(x+4) + (x-1)3x+1)(x-2)]/[(3x+1)(x-2)(x-5)(x+4)]
expand the top and collect like terms.
Good luck, since the result will certainly not be more simplified than the original.
To simplify the given expression, we need to simplify each fraction individually and then combine them by finding a common denominator, if necessary.
Let's simplify the first fraction, [2x / (3x^2 - 5x - 2)]:
1. Factorize the denominator: (3x^2 - 5x - 2)
We need to find two numbers whose product is equal to (3 * -2 = -6) and whose sum is equal to (-5).
The numbers that satisfy these conditions are -6 and 1.
Rewrite the middle term -5x as -6x + x:
(3x^2 - 6x + x - 2)
Group the terms and factor by grouping:
[(3x^2 - 6x) + (x - 2)]
Factor out common terms from each group:
[3x(x - 2) + 1(x - 2)]
Combine like terms:
[(3x + 1)(x - 2)]
Now, the first fraction becomes:
[2x / (3x + 1)(x - 2)]
Moving on to the second fraction, [(x-1) / (x^2 - x - 20)]:
2. Factorize the denominator: (x^2 - x - 20)
We need to find two numbers whose product is equal to (-20) and whose sum is equal to (-1).
The numbers that satisfy these conditions are -5 and 4.
Rewrite the middle term -x as -5x + 4x:
(x^2 - 5x + 4x - 20)
Group the terms and factor by grouping:
[(x^2 - 5x) + (4x - 20)]
Factor out common terms from each group:
[x(x - 5) + 4(x - 5)]
Combine like terms:
[(x + 4)(x - 5)]
Now, the second fraction becomes:
[(x - 1) / (x + 4)(x - 5)]
To combine the fractions, we need to find a common denominator. Since the denominators are already factored, the common denominator is simply the product of the two denominators:
Common denominator = (3x + 1)(x - 2)(x + 4)(x - 5)
Now, we can rewrite the expression with the common denominator:
[2x / (3x + 1)(x - 2)] + [(x - 1) / (x + 4)(x - 5)]
To add the fractions, we multiply the numerator of each fraction by the other fractions' denominator, and then combine the numerators over the common denominator:
[(2x)(x + 4)(x - 5) + (x - 1)(3x + 1)(x - 2)] / [(3x + 1)(x - 2)(x + 4)(x - 5)]
At this point, the expression is simplified as much as possible.