1. **State the problem:** Solve the quadratic equation $x^2 = 2x - 12$ for $x$. 2. **Rewrite the equation:** Move all terms to one side to set the equation to zero: $$x^2 - 2x + 12 = 0$$ 3. **Identify coefficients:** The quadratic is in the form $ax^2 + bx + c = 0$ where $a=1$, $b=-2$, and $c=12$. 4. **Use the quadratic formula:** $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$ 5. **Calculate the discriminant:** $$\Delta = b^2 - 4ac = (-2)^2 - 4 \times 1 \times 12 = 4 - 48 = -44$$ 6. **Interpret the discriminant:** Since $\Delta < 0$, there are no real solutions; solutions are complex. 7. **Find the complex solutions:** $$x = \frac{-(-2) \pm \sqrt{-44}}{2 \times 1} = \frac{2 \pm \sqrt{-44}}{2} = \frac{2 \pm \sqrt{44}i}{2}$$ 8. **Simplify $\sqrt{44}$:** $$\sqrt{44} = \sqrt{4 \times 11} = 2\sqrt{11}$$ 9. **Final solutions:** $$x = \frac{2 \pm 2\sqrt{11}i}{2} = 1 \pm \sqrt{11}i$$ **Answer:** The solutions are $x = 1 + \sqrt{11}i$ and $x = 1 - \sqrt{11}i$.