1. The problem is to write a complex equation and then provide its solution separately. 2. Let's consider the complex equation $$z^2 + (3 - 4i)z + (5 + 2i) = 0$$ where $z$ is a complex number. 3. To solve this quadratic equation in complex numbers, we use the quadratic formula: $$z = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$ where $a=1$, $b=3 - 4i$, and $c=5 + 2i$. 4. Calculate the discriminant: $$b^2 - 4ac = (3 - 4i)^2 - 4 \times 1 \times (5 + 2i)$$ 5. Compute $(3 - 4i)^2$: $$(3 - 4i)^2 = 3^2 - 2 \times 3 \times 4i + (4i)^2 = 9 - 24i + (-16) = -7 - 24i$$ 6. Compute $4ac$: $$4 \times 1 \times (5 + 2i) = 20 + 8i$$ 7. Calculate the discriminant: $$-7 - 24i - (20 + 8i) = -7 - 24i - 20 - 8i = -27 - 32i$$ 8. Now, find the square root of the discriminant $\sqrt{-27 - 32i}$. 9. Let $\sqrt{-27 - 32i} = x + yi$ where $x,y$ are real numbers. 10. Then: $$(x + yi)^2 = x^2 - y^2 + 2xyi = -27 - 32i$$ 11. Equate real and imaginary parts: $$x^2 - y^2 = -27$$ $$2xy = -32$$ 12. From $2xy = -32$, we get $xy = -16$. 13. Express $y = -16/x$ and substitute into the first equation: $$x^2 - \left(-\frac{16}{x}\right)^2 = -27$$ $$x^2 - \frac{256}{x^2} = -27$$ 14. Multiply both sides by $x^2$: $$x^4 + 27x^2 - 256 = 0$$ 15. Let $u = x^2$, then: $$u^2 + 27u - 256 = 0$$ 16. Solve quadratic in $u$: $$u = \frac{-27 \pm \sqrt{27^2 + 4 \times 256}}{2} = \frac{-27 \pm \sqrt{729 + 1024}}{2} = \frac{-27 \pm \sqrt{1753}}{2}$$ 17. Since $u = x^2$ must be positive, take the positive root: $$u = \frac{-27 + 41.87}{2} = 7.435$$ 18. Then $x = \pm \sqrt{7.435} = \pm 2.726$. 19. Find $y$: $$y = -\frac{16}{x} = -\frac{16}{2.726} = -5.87$$ 20. So, $\sqrt{-27 - 32i} = 2.726 - 5.87i$ or $-2.726 + 5.87i$. 21. Substitute back into quadratic formula: $$z = \frac{-(3 - 4i) \pm (2.726 - 5.87i)}{2}$$ 22. Calculate $-(3 - 4i) = -3 + 4i$. 23. Two solutions: $$z_1 = \frac{-3 + 4i + 2.726 - 5.87i}{2} = \frac{-0.274 - 1.87i}{2} = -0.137 - 0.935i$$ $$z_2 = \frac{-3 + 4i - 2.726 + 5.87i}{2} = \frac{-5.726 + 9.87i}{2} = -2.863 + 4.935i$$ Final answer: $$z_1 = -0.137 - 0.935i$$ $$z_2 = -2.863 + 4.935i$$