Solve: \( \frac{x^{2}-x-6}{x^{2}}=\frac{x-6}{2 x}+\frac{2 x+12}{x} \) After multiplying each side of the equation by the LCD and simplifying, the resulting equation is

To solve the equation \( \frac{x^{2}-x-6}{x^{2}}=\frac{x-6}{2 x}+\frac{2 x+12}{x} \), we first find the least common denominator (LCD), which is \( 2x^2 \). Multiplying both sides by \( 2x^2 \) gives us: \[ 2x^2 \cdot \frac{x^{2}-x-6}{x^{2}} = 2x^2 \cdot \frac{x-6}{2x} + 2x^2 \cdot \frac{2x+12}{x} \] This simplifies to: \[ 2(x^{2}-x-6) = x(x-6) + 2(2x+12) \] Expanding both sides, we get: \[ 2x^{2} - 2x - 12 = x^2 - 6x + 4x + 24 \] Further simplification leads to: \[ 2x^{2} - 2x - 12 = x^2 - 2x + 24 \] Rearranging gives: \[ 2x^2 - 2x - 12 - x^2 + 2x - 24 = 0 \] This results in: \[ x^2 - 36 = 0 \] So the simplified equation is: \[ x^2 - 36 = 0 \] Which factors to: \[ (x - 6)(x + 6) = 0 \] Thus, the solutions are \( x = 6 \) and \( x = -6 \).