1. **State the problem:** Simplify or analyze the quadratic expression $3x^2 - 4x + 1$. 2. **Recall the quadratic formula:** For any quadratic equation $ax^2 + bx + c = 0$, the solutions are given by $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$ where $a=3$, $b=-4$, and $c=1$. 3. **Calculate the discriminant:** $$\Delta = b^2 - 4ac = (-4)^2 - 4 \times 3 \times 1 = 16 - 12 = 4$$ 4. **Find the roots:** $$x = \frac{-(-4) \pm \sqrt{4}}{2 \times 3} = \frac{4 \pm 2}{6}$$ 5. **Evaluate each root:** - For the plus sign: $$x = \frac{4 + 2}{6} = \frac{6}{6} = 1$$ - For the minus sign: $$x = \frac{4 - 2}{6} = \frac{2}{6} = \frac{1}{3}$$ 6. **Final answer:** The roots of the quadratic $3x^2 - 4x + 1$ are $$x = 1 \quad \text{and} \quad x = \frac{1}{3}$$