Question

Solve the equation. Check the solution.
a/(a^2 - 16) + 2/(a-4) = 2/(a+4)

-16, -4
4
-16, 4
-16

which is the correct answer of the four above

Answer 1

a/(a^2 - 16) + 2/(a-4) = 2/(a+4)
a / [ ( a-4)(a+4)] + 2 (a+4) / [ ( a-4)(a+4)] = 2 (a-4) / [ ( a-4)(a+4)]
a + 2 a + 8 = 2 a - 8
3 a + 8 = 2 a - 8
a = -16

Answer 2

The correct answer is -16, 4.

To solve the equation, we first need to find a common denominator for all the fractions. The common denominator for a/(a^2 - 16), 2/(a-4), and 2/(a+4) is (a-4)(a+4)(a+4), which simplifies to (a-4)(a^2 - 16).

Multiplying each term by the common denominator, we get:

a + 2(a+4) = 2(a^2 - 16)
a + 2a + 8 = 2a^2 - 32
3a + 8 = 2a^2 - 32
0 = 2a^2 - 3a - 40
0 = (2a + 5)(a - 8)

Setting each factor to zero, we get:

2a + 5 = 0
2a = -5
a = -5/2

a - 8 = 0
a = 8

Therefore, the solutions are a = -5/2 and a = 8.

Checking the solutions by substituting them back into the original equation:

For a = -5/2:
-5/2/((-5/2)^2 - 16) + 2/(-5/2 - 4) = 2/(-5/2 + 4)
-5/2/(6.25 - 16) + 2/(-6.5) = 2/(-0.5)
-5/2/(-9.75) -4/13 = -4
-2/13 - 4/13 = -4
-6/13 = -4
This equation is not satisfied.

For a = 8:
8/(8^2 - 16) + 2/(8-4) = 2/(8+4)
8/(64 - 16) + 2/(4) = 2/(12)
8/48 + 2/4 = 2/12
2/12 + 2/4 = 2/12
1/6 + 1/2 = 1/6
1/3 = 1/6
This equation is not satisfied.

Therefore, there is no solution to this equation.

Answer 3

The correct solution is:

a = -16

Let's substitute a = -16 back into the original equation to check the solution:

-16 / [ (-16)^2 - 16] + 2 / (-16-4) = 2 / (-16+4)
-16 / [256 - 16] + 2 / -20 = 2 / -12
-16 / 240 - 1 / 10 = - 1 / 6
- 4 / 60 - 6 / 60 = - 10 / 60
-10 / 60 = -10 / 60

Therefore, the solution a = -16 is valid.