Use the discriminant to tell whether the solutions of this equation are real or complex \[ 3 x^{2}-4 x+10=0 \]

To determine the nature of the solutions of the quadratic equation \( 3x^2 - 4x + 10 = 0 \), we start by calculating the discriminant \( D \), which is given by the formula: \[ D = b^2 - 4ac \] Here, \( a = 3 \), \( b = -4 \), and \( c = 10 \). Plugging these values into the formula: \[ D = (-4)^2 - 4 \cdot 3 \cdot 10 = 16 - 120 = -104 \] Since the discriminant \( D \) is negative (\( -104 < 0 \)), it indicates that the solutions to the equation are complex. Therefore, the quadratic does not cross the x-axis and the solutions will be of the form \( a + bi \) where \( b \neq 0 \).