Solve the following equation. You must show all work to receive full credit.
a/a^2−25+2/a+5=2/a−5
First, find a common denominator for all fractions on each side of the equation.
a/(a^2−25) + 2/(a+5) = 2/(a−5)
The common denominator for all fractions in this equation is (a−5)(a+5)(a).
Now, rewrite each fraction with the common denominator:
a(a−5)(a)/((a−5)(a+5)(a)) + 2(a−5)/((a−5)(a+5)(a)) = 2(a+5)/((a−5)(a+5)(a))
Combine the fractions on the left side:
[a(a^2-5a) + 2(a−5)(a)] / (a−5)(a+5)(a) = 2(a+5)/((a−5)(a+5)(a))
Expand and simplify the numerator on the left side:
(a^3 - 5a^2 + 2a^2 - 10a) / (a−5)(a+5)(a) = 2(a+5)/((a−5)(a+5)(a))
Combine like terms:
(a^3 - 3a^2 - 10a) / (a−5)(a+5)(a) = 2(a+5)/((a−5)(a+5)(a))
Now, multiply both sides by the common denominator to get rid of the fractions:
a^3 - 3a^2 - 10a = 2(a+5)
Expand:
a^3 - 3a^2 - 10a = 2a + 10
Subtract 2a and 10 from both sides:
a^3 - 3a^2 - 10a - 2a - 10 = 0
Combine like terms:
a^3 - 3a^2 - 12a - 10 = 0
This is now a cubic equation. To solve it, you can use numerical methods or factorization.