Question:

The roots of the equation $\displaystyle 3x^2 - 3kx + k - 6 = 0$ are $\displaystyle \alpha$ and $\displaystyle \beta$. If $\displaystyle \alpha^2 + \beta^2$ = $\displaystyle \dfrac{20}{3}$, find the possible values of $\displaystyle k$ .

My workings:

$\displaystyle \alpha + \beta =$ $\displaystyle \dfrac{-b}{a}$ $\displaystyle = \dfrac{3k}{3} = k$

$\displaystyle \alpha\beta = $ $\displaystyle \dfrac{c}{a}$ $\displaystyle = \dfrac{k-6}{3}$

Can anyone facilitate in solving this problem?