Complex Solutions 411775

1. **State the problem:** Find all complex solutions of the quadratic equation $$3x^2 + 4x + 5 = 0.$$\n\n2. **Recall the quadratic formula:** For any quadratic equation $$ax^2 + bx + c = 0,$$ the solutions are given by $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}.$$\n\n3. **Identify coefficients:** Here, $$a = 3,$$ $$b = 4,$$ and $$c = 5.$$\n\n4. **Calculate the discriminant:** $$\Delta = b^2 - 4ac = 4^2 - 4 \times 3 \times 5 = 16 - 60 = -44.$$\n\n5. Since $$\Delta < 0,$$ the solutions are complex numbers.\n\n6. **Apply the quadratic formula:** $$x = \frac{-4 \pm \sqrt{-44}}{2 \times 3} = \frac{-4 \pm \sqrt{-1 \times 44}}{6} = \frac{-4 \pm i\sqrt{44}}{6}.$$\n\n7. **Simplify the square root:** $$\sqrt{44} = \sqrt{4 \times 11} = 2\sqrt{11}.$$\n\n8. **Substitute back:** $$x = \frac{-4 \pm 2i\sqrt{11}}{6}.$$\n\n9. **Simplify the fraction:** $$x = \frac{\cancel{2}(-2 \pm i\sqrt{11})}{\cancel{2} \times 3} = \frac{-2 \pm i\sqrt{11}}{3}.$$\n\n**Final answer:** $$x = \frac{-2 + i\sqrt{11}}{3}, \quad x = \frac{-2 - i\sqrt{11}}{3}.$$