Solve for x using the quadratic formula in the equation: \( 3x^{2} - 6x + 2 = 0 \)

To solve for \( x \) using the quadratic formula, we start with the equation \( 3x^{2} - 6x + 2 = 0 \). We can identify \( a = 3 \), \( b = -6 \), and \( c = 2 \). The quadratic formula is given by: \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \] Now, let's calculate the discriminant \( b^2 - 4ac \): \[ b^2 - 4ac = (-6)^2 - 4(3)(2) = 36 - 24 = 12 \] Next, we apply the quadratic formula: \[ x = \frac{-(-6) \pm \sqrt{12}}{2(3)} = \frac{6 \pm 2\sqrt{3}}{6} \] This simplifies to: \[ x = 1 \pm \frac{\sqrt{3}}{3} \] Thus, the solutions for \( x \) are: \[ x = 1 + \frac{\sqrt{3}}{3} \quad \text{and} \quad x = 1 - \frac{\sqrt{3}}{3} \] So, the roots are approximately \( x \approx 1.577 \) and \( x \approx 0.423 \). Enjoy exploring these solutions!