3x²-6x+2=0

To solve the quadratic equation \(3x^2 - 6x + 2 = 0\), we can use the quadratic formula:

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

In this equation, the coefficients are:

• \(a = 3\)
• \(b = -6\)
• \(c = 2\)

Now, we can calculate the discriminant (\(b^2 - 4ac\)):

\[ b^2 = (-6)^2 = 36 \] \[ 4ac = 4 \cdot 3 \cdot 2 = 24 \] \[ b^2 - 4ac = 36 - 24 = 12 \]

Since the discriminant is positive, we will have two real and distinct solutions.

Now, substituting into the quadratic formula:

\[ x = \frac{-(-6) \pm \sqrt{12}}{2 \cdot 3} \] \[ x = \frac{6 \pm \sqrt{12}}{6} \]

To simplify \(\sqrt{12}\):

\[ \sqrt{12} = \sqrt{4 \cdot 3} = 2\sqrt{3} \]

Substituting this back into our equation gives:

\[ x = \frac{6 \pm 2\sqrt{3}}{6} \]

Now simplify further:

\[ x = 1 \pm \frac{\sqrt{3}}{3} \]

So the solutions are:

\[ x = 1 + \frac{\sqrt{3}}{3} \quad \text{and} \quad x = 1 - \frac{\sqrt{3}}{3} \]

These are the two solutions to the quadratic equation \(3x^2 - 6x + 2 = 0\).