Help with this complex fraction please...
(a+2/a^2+a-2) - (3a+1/a^2+2a-3)
LCM is (a-1) (a+2) (a+3) right? then i get stuck
so now change each fraction to have that LCD
[ (a+2)(a+3) - (3a+1)(a+2) ]/[(a-1)(a+2)(a+3)]
expand the top and collect like terms, leave the bottom alone
=(a^2 + 5a + 6 - 3a^2 - 7a - 2)/[(a-1)(a+2)(a+3)]
= (-2a^2 - 2a + 4)/[(a-1)(a+2)(a+3)]
= -2(a^2 + a - 2)/[(a-1)(a+2)(a+3)]
= -2(a+2)(a-1)/[(a-1)(a+2)(a+3)]
= -2/(a+3)
Thanks!
To simplify complex fractions, we need to find a common denominator for all the fractions involved. However, in this case, we cannot use the LCM (Least Common Multiple) of the denominators (a-1), (a+2), (a+3) because it does not match the denominators in the given expression.
To simplify the expression, we should first factorize the denominators of each fraction:
Denominator of the first fraction: (a^2 + a - 2) = (a + 2)(a - 1)
Denominator of the second fraction: (a^2 + 2a - 3) = (a + 3)(a - 1)
Now, we have the correct factored denominators and can rewrite the expression:
(a+2)/(a+2)(a-1) - (3a+1)/(a+3)(a-1)
Next, we can multiply each term by the appropriate factor to eliminate the denominators:
[(a+2)/(a+2)(a-1)] * [(a+3)(a-1)/(a+3)(a-1)] - [(3a+1)/(a+3)(a-1)] * [(a+2)(a-1)/(a+2)(a-1)]
Simplifying further:
[(a+2)(a+3)(a-1) - (3a+1)(a+2)(a-1)] / [(a+2)(a-1)(a+3)]
Expanding the numerator:
[(a^2 + 5a + 6)(a-1) - (3a^2 + 7a + 2)(a-1)] / [(a+2)(a-1)(a+3)]
(a^3 + 5a^2 + 6a - a^2 - 5a - 6 - 3a^2 - 7a - 2a + 1) / [(a+2)(a-1)(a+3)]
Combining like terms:
(a^3 + a^2 - 6) / [(a+2)(a-1)(a+3)]
So, the simplified expression is:
(a^3 + a^2 - 6) / [(a+2)(a-1)(a+3)]