1. **State the problem:** We want to find the values of $a$ for which both roots of the quadratic function $f(x) = x^2 + ax + 1$ are greater than 0. 2. **Recall the quadratic formula:** The roots of $x^2 + ax + 1 = 0$ are given by $$x = \frac{-a \pm \sqrt{a^2 - 4}}{2}$$ 3. **Conditions for roots to be real and positive:** - The discriminant must be non-negative: $a^2 - 4 \geq 0$. - Both roots must be greater than 0. 4. **Analyze the discriminant:** $$a^2 - 4 \geq 0 \implies |a| \geq 2$$ 5. **Sum and product of roots:** For $x^2 + ax + 1 = 0$, sum of roots $S = -a$ and product of roots $P = 1$. 6. **Both roots positive means:** - $P = 1 > 0$ (already positive) - $S = -a > 0 \implies a < 0$ 7. **Combine conditions:** - $|a| \geq 2$ and $a < 0$ means $a \leq -2$ 8. **Conclusion:** Both roots are positive if and only if $$a \leq -2$$