Solve the equation by completing the square: \( x^{2}-12 x+56=3 \) \( \begin{array}{l}-6-\sqrt{17} i,-6+\sqrt{17} i \\ -6+17 i,-6-17 i \\ 6-17 i, 6+17 i \\ 0-\sqrt{17} i, 6+\sqrt{17} i \\ -17-\sqrt{3} i,-17+\sqrt{3} i\end{array} \)

To solve the equation \( x^{2}-12 x+56=3 \) by completing the square, we will follow these steps: 1. **Rearrange the equation**: Move all terms to one side of the equation. \[ x^{2} - 12x + 56 - 3 = 0 \] This simplifies to: \[ x^{2} - 12x + 53 = 0 \] 2. **Complete the square**: We need to rewrite the quadratic in the form \( (x - p)^2 = q \). - Take the coefficient of \( x \), which is \(-12\), divide it by 2 to get \(-6\), and then square it to get \(36\). - Add and subtract \(36\) inside the equation: \[ x^{2} - 12x + 36 - 36 + 53 = 0 \] This can be rewritten as: \[ (x - 6)^{2} - 36 + 53 = 0 \] Simplifying gives: \[ (x - 6)^{2} + 17 = 0 \] 3. **Isolate the square**: \[ (x - 6)^{2} = -17 \] 4. **Take the square root of both sides**: \[ x - 6 = \pm \sqrt{-17} \] This can be expressed as: \[ x - 6 = \pm i\sqrt{17} \] 5. **Solve for \( x \)**: \[ x = 6 \pm i\sqrt{17} \] Thus, the solutions to the equation \( x^{2}-12 x+56=3 \) are: \[ x = 6 + i\sqrt{17}, \quad x = 6 - i\sqrt{17} \] Now, let's check the provided options: - The correct solutions are \( 6 + \sqrt{17} i \) and \( 6 - \sqrt{17} i \). Therefore, the correct answer is: \[ 6 + \sqrt{17} i, \quad 6 - \sqrt{17} i \]