1. The problem is to solve the quadratic equation $$7x^2 + 2x + 8 = 0$$ using the quadratic formula. 2. The quadratic formula is given by: $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$ where $a$, $b$, and $c$ are coefficients from the quadratic equation $ax^2 + bx + c = 0$. 3. Identify the coefficients: - $a = 7$ - $b = 2$ - $c = 8$ 4. Calculate the discriminant: $$\Delta = b^2 - 4ac = 2^2 - 4 \times 7 \times 8 = 4 - 224 = -220$$ 5. Since the discriminant is negative ($\Delta = -220$), the solutions are complex (no real roots). 6. Calculate the roots using the quadratic formula: $$x = \frac{-2 \pm \sqrt{-220}}{2 \times 7} = \frac{-2 \pm \sqrt{220}i}{14}$$ 7. Simplify $\sqrt{220}$: $$\sqrt{220} = \sqrt{4 \times 55} = 2\sqrt{55}$$ 8. Substitute back: $$x = \frac{-2 \pm 2\sqrt{55}i}{14} = \frac{-2}{14} \pm \frac{2\sqrt{55}i}{14} = -\frac{1}{7} \pm \frac{\sqrt{55}}{7}i$$ Final answer: $$x = -\frac{1}{7} \pm \frac{\sqrt{55}}{7}i$$