3x²-5x+4=0

To solve the quadratic equation \( 3x^2 - 5x + 4 = 0 \), we can use the quadratic formula:

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

In this equation, \( a = 3 \), \( b = -5 \), and \( c = 4 \).

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

\[ b^2 - 4ac = (-5)^2 - 4 \cdot 3 \cdot 4 = 25 - 48 = -23 \]

Since the discriminant is negative (\(-23\)), this means that the equation does not have real solutions, but rather two complex solutions.

Now we can plug the values into the quadratic formula:

\[ x = \frac{-(-5) \pm \sqrt{-23}}{2 \cdot 3} = \frac{5 \pm \sqrt{23}i}{6} \]

Thus, the solutions to the equation \( 3x^2 - 5x + 4 = 0 \) are:

\[ x = \frac{5 + \sqrt{23}i}{6} \quad \text{and} \quad x = \frac{5 - \sqrt{23}i}{6} \]