Solve the equation. \[ \frac{5}{x^{2}}-\frac{54}{x}=11 \]

To solve the equation \[ \frac{5}{x^{2}} - \frac{54}{x} = 11, \] first, let's eliminate the fractions by multiplying both sides by \(x^2\) (assuming \(x \neq 0\)): \[ 5 - 54x = 11x^2. \] Rearranging the equation gives us: \[ 11x^2 + 54x - 5 = 0. \] Now, we can solve this quadratic equation using the quadratic formula, where \(a = 11\), \(b = 54\), and \(c = -5\): \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}. \] Calculating the discriminant: \[ b^2 - 4ac = 54^2 - 4 \cdot 11 \cdot (-5) = 2916 + 220 = 3136. \] Now, taking the square root: \[ \sqrt{3136} = 56. \] Now we can substitute back into the quadratic formula: \[ x = \frac{-54 \pm 56}{2 \cdot 11}. \] Calculating the two possible values for \(x\): 1. \(x = \frac{-54 + 56}{22} = \frac{2}{22} = \frac{1}{11}\). 2. \(x = \frac{-54 - 56}{22} = \frac{-110}{22} = -5\). Thus, the solutions to the equation are: \[ x = \frac{1}{11} \quad \text{and} \quad x = -5. \] In conclusion, the solutions to the original equation are \(x = \frac{1}{11}\) and \(x = -5\).