Question
solve the equation \frac{a}{a^{2}-16}+\frac{2}{a-4}=\frac{2}{a+4}
Answers
GPT 3.5
To solve this equation, we first find a common denominator. The common denominator for the fractions on the left side is (a+4)(a-4).
Rewriting the equation with the common denominator, we get:
\( \frac{a(a+4)}{(a+4)(a-4)} + \frac{2(a-4)}{(a+4)(a-4)} = \frac{2}{a+4} \)
Now, we combine the fractions on the left side:
\( \frac{a(a+4) + 2(a-4)}{(a+4)(a-4)} = \frac{2}{a+4} \)
Expanding the numerator and simplifying, we get:
\( \frac{a^{2} + 4a + 2a - 8}{(a+4)(a-4)} = \frac{2}{a+4} \)
\( \frac{a^{2} + 6a - 8}{(a+4)(a-4)} = \frac{2}{a+4} \)
Multiplying both sides by (a+4), we get:
\( a^{2} + 6a - 8 = 2(a-4) \)
\( a^{2} + 6a - 8 = 2a - 8 \)
Subtracting 2a from both sides, we get:
\( a^{2} + 4a = 0 \)
\( a(a + 4) = 0 \)
So, the solutions to the equation are \( a = 0 \) or \( a = -4 \).
Rewriting the equation with the common denominator, we get:
\( \frac{a(a+4)}{(a+4)(a-4)} + \frac{2(a-4)}{(a+4)(a-4)} = \frac{2}{a+4} \)
Now, we combine the fractions on the left side:
\( \frac{a(a+4) + 2(a-4)}{(a+4)(a-4)} = \frac{2}{a+4} \)
Expanding the numerator and simplifying, we get:
\( \frac{a^{2} + 4a + 2a - 8}{(a+4)(a-4)} = \frac{2}{a+4} \)
\( \frac{a^{2} + 6a - 8}{(a+4)(a-4)} = \frac{2}{a+4} \)
Multiplying both sides by (a+4), we get:
\( a^{2} + 6a - 8 = 2(a-4) \)
\( a^{2} + 6a - 8 = 2a - 8 \)
Subtracting 2a from both sides, we get:
\( a^{2} + 4a = 0 \)
\( a(a + 4) = 0 \)
So, the solutions to the equation are \( a = 0 \) or \( a = -4 \).