The equation you provided is a quadratic equation:

$x^2 - 2x + 3 = 0$

To solve it, follow these steps:

1. Identify the coefficients from the quadratic equation $ax^2 + bx + c = 0$:

   • $a = 1$
   • $b = -2$
   • $c = 3$

2. Use the quadratic formula: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ Substituting the values: $x = \frac{-(-2) \pm \sqrt{(-2)^2 - 4(1)(3)}}{2(1)}$ Simplifying the terms: $x = \frac{2 \pm \sqrt{4 - 12}}{2}$ $x = \frac{2 \pm \sqrt{-8}}{2}$ $x = \frac{2 \pm 2i\sqrt{2}}{2}$

3. Simplify: $x = 1 \pm i\sqrt{2}$

The solutions are:

$x = 1 + i\sqrt{2} \quad \text{or} \quad x = 1 - i\sqrt{2}$ These are complex solutions.

Would you like more details on complex numbers or the quadratic formula?

Related questions:

1. What are complex numbers, and how do they work?
2. How do you complete the square for quadratic equations?
3. What is the geometric meaning of complex roots?
4. How can we factor quadratic equations with real solutions?
5. How does the discriminant $b^2 - 4ac$ determine the nature of the roots?

Tip: The discriminant in the quadratic formula helps to quickly determine whether the solutions are real or complex.