Questions: Use the quadratic formula to solve for (x). [3 x^2-6 x+2=0] Round your answer to the nearest hundredth. If there is more than one solution, separate them with commas.

Solution Steps

To solve the quadratic equation \(3x^2 - 6x + 2 = 0\) using the quadratic formula, we first identify the coefficients \(a = 3\), \(b = -6\), and \(c = 2\). The quadratic formula is given by:

\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]

We will calculate the discriminant \(b^2 - 4ac\) and then use it to find the two possible values for \(x\). Finally, we will round the solutions to the nearest hundredth.

For the quadratic equation \(3x^2 - 6x + 2 = 0\), we identify the coefficients as follows:

• \(a = 3\)
• \(b = -6\)
• \(c = 2\)

We calculate the discriminant using the formula \(D = b^2 - 4ac\): \[ D = (-6)^2 - 4 \cdot 3 \cdot 2 = 36 - 24 = 12 \]

Using the quadratic formula \(x = \frac{-b \pm \sqrt{D}}{2a}\), we find the two solutions: \[ x_1 = \frac{-(-6) + \sqrt{12}}{2 \cdot 3} = \frac{6 + 2\sqrt{3}}{6} = 1 + \frac{\sqrt{3}}{3} \] \[ x_2 = \frac{-(-6) - \sqrt{12}}{2 \cdot 3} = \frac{6 - 2\sqrt{3}}{6} = 1 - \frac{\sqrt{3}}{3} \]

Calculating the numerical values and rounding to the nearest hundredth: \[ x_1 \approx 1.58 \] \[ x_2 \approx 0.42 \]

Final Answer