To solve the quadratic equation $x^2 - 2x + 3 = 0$ by completing the square, follow these steps:

1. Rewrite the equation:
   Start with the given equation:
   $x^2 - 2x + 3 = 0$

2. Move the constant term to the other side:
   Subtract 3 from both sides: $x^2 - 2x = -3$

3. Complete the square:
   To complete the square, take half of the coefficient of $x$ (which is $-2$), square it, and add it to both sides: $\left(\frac{-2}{2}\right)^2 = 1$ Add 1 to both sides: $x^2 - 2x + 1 = -3 + 1$ $(x - 1)^2 = -2$

4. Solve for $x$:
   Now, take the square root of both sides: $x - 1 = \pm \sqrt{-2}$ Since $\sqrt{-2} = i\sqrt{2}$, where $i$ is the imaginary unit, we get: $x - 1 = \pm i\sqrt{2}$ Solve for $x$ by adding 1 to both sides: $x = 1 \pm i\sqrt{2}$

Thus, the solutions are: $x = 1 + i\sqrt{2} \quad \text{or} \quad x = 1 - i\sqrt{2}$

Would you like more details on any step or have any questions?

Here are 5 related questions you might explore:

1. How do you solve quadratic equations using the quadratic formula?
2. What is the discriminant of a quadratic equation, and what does it tell us?
3. Can all quadratic equations be solved by completing the square?
4. How do you perform arithmetic with complex numbers like $1 \pm i\sqrt{2}$?
5. What is the geometric interpretation of the roots of a quadratic equation?

Tip: Completing the square is also useful for converting quadratic equations into vertex form, which makes it easier to graph.