Question

solve the equation \frac{a}{a^{2}-16}+\frac{2}{a-4}=\frac{2}{a+4}
-16, -4
-16
-16, 4
4

Answer 1

First, we need to find a common denominator for the fractions on the left side of the equation. The common denominator is (a+4)(a-4).

Then, rewrite the equation using the common denominator:
a/(a^2-16) * (a+4)/(a+4) + 2/(a-4) * (a-4)/(a-4) = 2/(a+4)

This simplifies to:
a(a+4)/(a^2-16) + 2(a-4)/(a^2-16) = 2/(a+4)

Combine the fractions on the left side:
[a(a+4) + 2(a-4)] / (a^2-16) = 2/(a+4)

Expand and simplify the numerator:
(a^2 + 4a + 2a - 8) / (a^2-16) = 2/(a+4)
(a^2 + 6a - 8) / (a^2-16) = 2/(a+4)

Now, cross multiply:
(a^2 + 6a - 8)(a+4) = 2(a^2-16)
a^3 + 4a^2 + 6a^2 + 24a - 8a - 32 = 2a^2 - 32
a^3 + 10a^2 + 16a - 32 = 2a^2 - 32

Combine like terms:
a^3 + 8a^2 + 16a = 0

Factor out an 'a' and solve for 'a':
a(a^2 + 8a + 16) = 0
a(a+4)^2 = 0
a = 0 or a = -4

Therefore, the solutions to the equation are a = 0 and a = -4.