1. **State the problem:** Solve the quadratic equation $-3x^2 + 2x + 1 = 0$. 2. **Formula used:** 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 coefficients:** Here, $a = -3$, $b = 2$, and $c = 1$. 4. **Calculate the discriminant:** $$\Delta = b^2 - 4ac = 2^2 - 4(-3)(1) = 4 + 12 = 16$$ 5. **Apply the quadratic formula:** $$x = \frac{-2 \pm \sqrt{16}}{2(-3)} = \frac{-2 \pm 4}{-6}$$ 6. **Find the two solutions:** - For the plus sign: $$x = \frac{-2 + 4}{-6} = \frac{2}{-6} = -\frac{1}{3}$$ - For the minus sign: $$x = \frac{-2 - 4}{-6} = \frac{-6}{-6} = 1$$ 7. **Final answer:** The solutions to the equation are $$x = -\frac{1}{3} \quad \text{and} \quad x = 1$$