If the roots of the equation $x^{3} + a x^{2} + b x + c = 0$ are in A.P., then $2 a^{3} - 9 a b =$ _____.

9 c

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18 c

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27 c

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- 27 c

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Solution

The correct option is B $- 27 c$
Let the roots of the given equation be $s - t, s$ and $s + t$ .
We can thus write the following conclusions:
$s - t + s + s + t = - a$
$\Rightarrow - 3 s = a$
$2. (s - t) s + s (s + t) + (s - t) (s + t) = b$
$\Rightarrow s^{2} - s t + s^{2} + s t + s^{2} - t^{2} = 3 s^{2} - t^{2} = b$
$\Rightarrow 3. (s - t) s (s + t) = - c$
$\Rightarrow - s^{3} + s t^{2} = c$
$∴ 2 a^{3} - 9 a b = 2 (- 3 s)^{3} - 9 (- 3 s) (3 s^{2} - t^{2}) = - 54 s^{3} + 81 s^{3} - 27 s t^{2}$
$= 27 s^{3} - 27 s t^{2}$
$= - 27 c$

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