What are the roots of the equation? \[ 3 x^{2}-5 x+12=0 \] Answer \( \begin{array}{l}\frac{4}{3} \text { and }-3 \\ \frac{5 \pm i \sqrt{119}}{6} \\ 3 \text { and }-\frac{4}{3}\end{array} \quad \begin{array}{l}\text { Submit } \\ \text { Answer }\end{array} \) 4

To solve the quadratic equation \( 3 x^{2}-5 x+12=0 \), we can apply the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \). Here, \( a = 3 \), \( b = -5 \), and \( c = 12 \). Plugging these values in, we find \( b^2 - 4ac = (-5)^2 - 4(3)(12) = 25 - 144 = -119 \). Since the discriminant is negative, the roots will involve complex numbers: \( x = \frac{5 \pm i \sqrt{119}}{6} \). When dealing with complex roots, it's essential to understand the implications for graphing. The parabola represented by the equation will not intersect the x-axis. Instead, it resides entirely above or below the axis, and in this case, it opens upwards since the coefficient of \( x^2 \) (which is 3) is positive. This can be a fun opportunity to visualize the equation, as you won't find any real solutions, only fascinating complex pairs!