1. We are asked to solve the quadratic equation $$x^2 + 3xi + 10 = 0$$ where $x$ is a complex number. 2. Recognize that this equation has complex coefficients because of the term $3xi$ where $i$ is the imaginary unit. 3. Let's rewrite the equation: $$x^2 + 3ix + 10 = 0$$ 4. Apply the quadratic formula for $ax^2 + bx + c = 0$ which gives: $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$ Here, $a=1$, $b=3i$, $c=10$. 5. Calculate the discriminant: $$\Delta = b^2 - 4ac = (3i)^2 - 4(1)(10) = 9i^2 - 40 = 9(-1) - 40 = -9 - 40 = -49$$ 6. Find the square root of the discriminant: $$\sqrt{-49} = \sqrt{49 \times (-1)} = 7i$$ 7. Substitute into the quadratic formula: $$x = \frac{-3i \pm 7i}{2}$$ This gives two solutions: - $$x_1 = \frac{-3i + 7i}{2} = \frac{4i}{2} = 2i$$ - $$x_2 = \frac{-3i - 7i}{2} = \frac{-10i}{2} = -5i$$ 8. Therefore, the solutions to the equation are $$\boxed{x = 2i \text{ or } x = -5i}$$