If $\displaystyle \alpha , \beta $ are roots of the equation $\displaystyle 3x^{2}-11x+19=0 $, then it is possible to find the value of $\displaystyle \alpha^{2}+\beta^{2}$.

Using the quadratic formula properties for 3x squared - 11x + 19 = 0, the sum of the roots alpha and beta is 11/3 and their product is 19/3. The identity alpha squared + beta squared equals (alpha + beta) squared minus 2*alpha*beta applies here. Substituting the values yields (11/3) squared - 2(19/3) = 121/9 - 114/9 = 7/9. Therefore, the value can be found directly from the coefficients, validating both the assertion and its underlying reason.

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