1. **State the problem:** We are given two numbers $\alpha$ and $\beta$ such that their sum is $\alpha + \beta = -6$ and their product is $\alpha \beta = 1$. 2. **Interpretation:** These conditions describe the roots of a quadratic equation in the form $$x^2 - (\alpha + \beta)x + \alpha \beta = 0.$$ Substituting, the equation becomes $$x^2 - (-6)x + 1 = 0,$$ which simplifies to $$x^2 + 6x + 1 = 0.$$ 3. **Solve the quadratic equation:** Use the quadratic formula $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$ where $a=1$, $b=6$, and $c=1$. 4. **Calculate the discriminant:** $$\Delta = b^2 - 4ac = 6^2 - 4(1)(1) = 36 - 4 = 32.$$ 5. **Find the roots:** $$x = \frac{-6 \pm \sqrt{32}}{2} = \frac{-6 \pm 4\sqrt{2}}{2} = -3 \pm 2\sqrt{2}.$$ 6. **Conclusion:** The values of $\alpha$ and $\beta$ that satisfy the given conditions are $$\alpha = -3 + 2\sqrt{2}$$ and $$\beta = -3 - 2\sqrt{2}.$$