1. **Problem:** Solve the quadratic equation $x^2 + 3x + 9 = 0$. 2. **Formula:** Use the quadratic formula: $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$ where $a=1$, $b=3$, and $c=9$. 3. **Calculate the discriminant:** $$\Delta = b^2 - 4ac = 3^2 - 4 \times 1 \times 9 = 9 - 36 = -27$$ 4. **Interpretation:** Since $\Delta < 0$, the roots are complex (no real roots). 5. **Find the roots:** $$x = \frac{-3 \pm \sqrt{-27}}{2} = \frac{-3 \pm i\sqrt{27}}{2} = \frac{-3 \pm 3i\sqrt{3}}{2}$$ 6. **Final answer:** $$x = \frac{-3}{2} \pm \frac{3i\sqrt{3}}{2}$$ --- 1. **Problem:** Solve the quadratic equation $\sqrt{2} x^2 + x + \sqrt{2} = 0$. 2. **Formula:** Use the quadratic formula: $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$ where $a=\sqrt{2}$, $b=1$, and $c=\sqrt{2}$. 3. **Calculate the discriminant:** $$\Delta = b^2 - 4ac = 1^2 - 4 \times \sqrt{2} \times \sqrt{2} = 1 - 4 \times 2 = 1 - 8 = -7$$ 4. **Interpretation:** Since $\Delta < 0$, the roots are complex. 5. **Find the roots:** $$x = \frac{-1 \pm \sqrt{-7}}{2\sqrt{2}} = \frac{-1 \pm i\sqrt{7}}{2\sqrt{2}}$$ 6. **Final answer:** $$x = \frac{-1}{2\sqrt{2}} \pm \frac{i\sqrt{7}}{2\sqrt{2}}$$