Question

\[3 x^2 - 4x + \frac{20}{3} = 0\]

Answer 1

Given:  

\[3 x^2 - 4 x + \frac{20}{3} = 0\]

Comparing the given equation with the general form of the quadratic equation 

\[a x^2 + bx = c = 0\] , we get

\[a = 3, b = - 4\] and \[c = \frac{20}{3}\].

Substituting these values in

\[\alpha = \frac{- b + \sqrt{b^2 - 4ac}}{2a}\] and \[\beta = \frac{- b - \sqrt{b^2 - 4ac}}{2a}\], we get:

\[\Rightarrow \alpha = \frac{4 + \sqrt{16 - 4 \times 3 \times \frac{20}{3}}}{6}\] and   \[\beta = \frac{4 - \sqrt{16 - 4 \times 3 \times \frac{20}{3}}}{6}\]

\[\Rightarrow \alpha = \frac{4 + \sqrt{- 64}}{6}\]  and   \[\beta = \frac{4 - \sqrt{- 64}}{6}\]

\[\Rightarrow \alpha = \frac{4 + 8i}{6}\] and \[\beta = \frac{4 - 8i}{6}\] 

\[\Rightarrow \alpha = \frac{2 + 4i}{3}\]  and \[\beta = \frac{2 - 4i}{3}\]

Hence, the roots of the equation are  

\[\frac{2 \pm 4i}{3}\].