1. **State the problem:** Solve the quadratic equation $$4a^2 + 4a + 1 = 0$$ using the quadratic formula. 2. **Recall the quadratic formula:** For an equation $$ax^2 + bx + c = 0$$, the solutions are given by $$a = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$ where $a$, $b$, and $c$ are coefficients from the equation. 3. **Identify coefficients:** Here, $a=4$, $b=4$, and $c=1$. 4. **Calculate the discriminant:** $$b^2 - 4ac = 4^2 - 4 \times 4 \times 1 = 16 - 16 = 0$$ 5. **Apply the quadratic formula:** $$a = \frac{-4 \pm \sqrt{0}}{2 \times 4} = \frac{-4 \pm 0}{8}$$ 6. **Simplify the expression:** $$a = \frac{-4}{8} = \frac{\cancel{4} \times (-1)}{\cancel{4} \times 2} = -\frac{1}{2}$$ 7. **Final answer:** The equation has one rational solution: $$a = -\frac{1}{2}$$