Solve the equation by completing the square: \( z^{2}-4 z+20=3 \)

To solve the equation \( z^{2} - 4z + 20 = 3 \) by completing the square, first, let's rewrite it in standard form: 1. Subtract 3 from both sides: \[ z^{2} - 4z + 17 = 0 \] 2. Now, we focus on the quadratic expression \( z^{2} - 4z \). To complete the square, we take half of the coefficient of \( z \), which is \(-4\), giving us \(-2\). We then square this value: \[ (-2)^2 = 4 \] 3. Add and subtract this squared value inside the equation: \[ z^{2} - 4z + 4 - 4 + 17 = 0 \] This simplifies to: \[ (z - 2)^{2} + 13 = 0 \] 4. Now, isolate the square term: \[ (z - 2)^{2} = -13 \] 5. Since we can't take the square root of a negative number in the real number system, we introduce imaginary numbers: \[ z - 2 = \pm i\sqrt{13} \] 6. Finally, solving for \( z \): \[ z = 2 \pm i\sqrt{13} \] Thus, the solutions to the equation are \( z = 2 + i\sqrt{13} \) and \( z = 2 - i\sqrt{13} \).