Solve the rational equation: \( 1-\frac{6}{x}=-\frac{8}{x^{2}} \) \( x=\square \)

To solve the rational equation: \[ 1 - \frac{6}{x} = -\frac{8}{x^{2}} \] **Step 1: Eliminate the Denominators** Multiply both sides of the equation by \( x^{2} \) to eliminate the denominators: \[ x^{2} \left(1 - \frac{6}{x}\right) = x^{2} \left(-\frac{8}{x^{2}}\right) \] Simplify both sides: \[ x^{2} - 6x = -8 \] **Step 2: Bring All Terms to One Side** Move all terms to the left side to set the equation to zero: \[ x^{2} - 6x + 8 = 0 \] **Step 3: Solve the Quadratic Equation** Factor the quadratic equation: \[ x^{2} - 6x + 8 = (x - 2)(x - 4) = 0 \] Set each factor equal to zero: \[ x - 2 = 0 \quad \text{or} \quad x - 4 = 0 \] So, the solutions are: \[ x = 2 \quad \text{and} \quad x = 4 \] **Step 4: Verify the Solutions** Ensure that neither solution makes any denominator in the original equation zero. Since \( x = 2 \) and \( x = 4 \) are both non-zero, they are valid solutions. **Final Answer:** \[ x = 2 \quad \text{or} \quad x = 4 \]