The given problem is a quadratic equation: $x^2 - 2x + 6 = 0$. To solve this, we will use the quadratic formula:

$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

Here:

• $a = 1$,
• $b = -2$,
• $c = 6$.

Substitute these into the formula:

$x = \frac{-(-2) \pm \sqrt{(-2)^2 - 4(1)(6)}}{2(1)}$ $x = \frac{2 \pm \sqrt{4 - 24}}{2}$ $x = \frac{2 \pm \sqrt{-20}}{2}$

Since the discriminant ($-20$) is negative, the solutions involve imaginary numbers:

$x = \frac{2 \pm \sqrt{-20}}{2}$ $x = \frac{2 \pm 2i\sqrt{5}}{2}$ $x = 1 \pm i\sqrt{5}$

Solutions:

$x = 1 + i\sqrt{5}, \quad x = 1 - i\sqrt{5}$

Now, here are your requested problems:

Problems Involving Quadratic Equations:

1. Solve $2x^2 - 3x - 2 = 0$ using the quadratic formula.
2. Solve $x^2 + 4x + 8 = 0$, which involves imaginary numbers.
3. Factorize $x^2 - 5x + 6 = 0$.
4. Solve $3x^2 + x - 4 = 0$ by completing the square.
5. Solve $x^2 + 2x + 10 = 0$ and express the roots in terms of imaginary numbers.

Tip:

Always check the discriminant $(b^2 - 4ac)$ first to determine whether the roots are real, repeated, or complex. Let me know which problems you'd like help solving or if you'd like further explanation!