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Sign of Quadratic Expression

Assume that ...

Question

Assume that $x_{1}$ and $x_{2}$ are roots of the equation $3 x^{2} - a x + 2 a - 1 = 0$ . Calculate $x_{1}^{3} + x_{2}^{3}$

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Solution

$3 x^{2} - a x + 2 a - 1 = 0$
Sum $= x_{1} + x_{2} = \frac{a}{3}$
Product $= x_{1} x_{2} = \frac{2 a - 1}{3}$
$x_{1}^{3} + x_{2}^{3} = (x_{1} + x_{2})^{3} - 3 x_{1} x_{2} (x_{1} + x_{2})$
$(\frac{a}{3})^{3} - 3 (\frac{2 a - 1}{3}) (\frac{a}{3})$
$\frac{a^{3}}{27} - \frac{2 a^{2} - a}{3}$
$= \frac{a^{3} - 18 a^{2} + 9 a}{27}$
$= \frac{a}{27} (a^{2} - 18 a + 9)$

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