If a and b are the roots of the quadratic equation ${2 x}^{2} - 6 x + 3 = 0$ , find the value of
$a^{3} + b^{3} - 3 a b (a^{2} + b^{2}) - 3 a b (a + b)$ .

-27

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-25

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None of these

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Solution

The correct option is A -27
Given equation is: $2 x^{2} - 6 x + 3 = 0$
$a$ and $b$ are roots of the given equation.
$∴ a + b = 3; a b = \frac{3}{2}$
Now, $a^{3} + b^{3} - 3 a b (a^{2} + b^{2}) - 3 a b (a + b)$
$= {(a + b)}^{3} - 3 a b (a + b) - 3 a b [{(a + b)}^{2} - 2 a b] - 3 a b (a + b)$
$= {(3)}^{3} - 3 \times \frac{3}{2} \times 3 - 3 \times \frac{3}{2} [3^{2} - 2 \times \frac{3}{2}] - 3 \times \frac{3}{2} \times 3$
$= 27 - \frac{27}{2} - \frac{9}{2} \times 6 - \frac{27}{2}$
$= - 27$

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