X-3X²+2=0

To solve the quadratic equation \( -3X^2 + X + 2 = 0 \), we can rearrange it into a standard form:

\[ 3X^2 - X - 2 = 0 \]

Now we can use the quadratic formula, which is given by:

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

Here, \( a = 3 \), \( b = -1 \), and \( c = -2 \).

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

\[ b^2 = (-1)^2 = 1 \] \[ 4ac = 4 \cdot 3 \cdot (-2) = -24 \] \[ b^2 - 4ac = 1 - (-24) = 1 + 24 = 25 \]

Now we can plug the values into the quadratic formula:

\[ X = \frac{-(-1) \pm \sqrt{25}}{2 \cdot 3} \] \[ X = \frac{1 \pm 5}{6} \]

This gives us two potential solutions:

1. When we take the positive root: \[ X = \frac{1 + 5}{6} = \frac{6}{6} = 1 \]

2. When we take the negative root: \[ X = \frac{1 - 5}{6} = \frac{-4}{6} = -\frac{2}{3} \]

Thus, the solutions to the equation \( -3X^2 + X + 2 = 0 \) are:

\[ X = 1 \quad \text{and} \quad X = -\frac{2}{3} \]