How do you solve #a/(2a+1) - (2a^2+5)/ (2a^2-5a-3) =3/(a-3)#?

Question

How do you solve #a/(2a+1) - (2a^2+5)/ (2a^2-5a-3)    =3/(a-3)#?

Eli Assouline · Accepted Answer

Given #a/(2a+1)-(2a^2+5)/(2a^2-5a-3) = 3/(a-3)# Noting that #2a^2-5a-3 = (2a+1)(a-3)#
we can clear the denominators from the given equation by multiplying by #2a^2-5a-3# #a(a-3) -(2a^2+5) = 3(2a+1)# #a^2 -3a -2a^2 -5 = 6a +3# #a^2+9a+8 =0# Factor the left side
#(a+1)(a+9) = 0# State the solutions
#a=-1# or #a=-9#

Kennedy Adams · Answer

undefined To solve the equation (a/(2a+1)) - ((2a^2+5)/ (2a^2-5a-3)) = 3/(a-3), we can follow these steps:

1. Simplify the expressions on both sides of the equation.
2. Find a common denominator for the fractions.
3. Combine the fractions on both sides of the equation.
4. Solve for the variable, a.

Here are the steps in detail:

1. Simplify the expressions:
   - The expression a/(2a+1) is already simplified.
   - For the expression (2a^2+5)/ (2a^2-5a-3), we cannot simplify it further.

2. Find a common denominator:
   - The denominators in the two fractions are (2a+1) and (2a^2-5a-3).
   - The common denominator is (2a+1)(2a^2-5a-3).

3. Combine the fractions:
   - Multiply the first fraction, a/(2a+1), by (2a^2-5a-3)/(2a^2-5a-3).
   - Multiply the second fraction, (2a^2+5)/ (2a^2-5a-3), by (2a+1)/(2a+1).
   - This will give us a common denominator for both fractions.

4. Solve for the variable, a:
   - After combining the fractions, we will have a new equation.
   - Solve this equation for the variable, a.
   - The solution will be the value(s) of a that satisfy the equation.

Please note that without the specific equation, it is not possible to provide the exact solution.