Questions: Solve for x. If there is more than one solution, separate the solutions with a comma. 5/(x+2) + x/(x-2) = (-7x-58)/(x^2-4) x=

Solution Steps

To solve the equation \(\frac{5}{x+2}+\frac{x}{x-2}=\frac{-7x-58}{x^{2}-4}\), we need to recognize that \(x^2 - 4\) can be factored into \((x+2)(x-2)\). This allows us to combine the fractions on the left-hand side over a common denominator. Once we have a single fraction on each side of the equation, we can set the numerators equal to each other and solve for \(x\).

We start with the equation:

\[ \frac{5}{x+2} + \frac{x}{x-2} = \frac{-7x - 58}{x^2 - 4} \]

Recognizing that \(x^2 - 4\) can be factored as \((x+2)(x-2)\), we can rewrite the equation with a common denominator.

Combining the fractions on the left-hand side gives us:

\[ \frac{5(x-2) + x(x+2)}{(x+2)(x-2)} = \frac{-7x - 58}{(x+2)(x-2)} \]

This simplifies to:

\[ 5(x-2) + x(x+2) = -7x - 58 \]

Expanding both sides leads to:

\[ 5x - 10 + x^2 + 2x = -7x - 58 \]

Combining like terms results in:

\[ x^2 + 14x + 48 = 0 \]

Factoring the quadratic gives us:

\[ (x + 6)(x + 8) = 0 \]

Thus, the solutions are:

\[ x = -6, \quad x = -8 \]

Final Answer