The possible value(s) of $a$ for which the equation $x^{2} + 2 x - (a^{3} - 3 a - 3) = 0$ has real roots is/are

- 1

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4

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2

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6

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Solution

The correct option is D $6$
$x^{2} + 2 x - (a^{3} - 3 a - 3) = 0$
For real roots,
$D \geq 0$
$\Rightarrow 4 + 4 (a^{3} - 3 a - 3) \geq 0 \Rightarrow 1 + a^{3} - 3 a - 3 \geq 0 \Rightarrow a^{3} - 3 a - 2 \geq 0 \Rightarrow (a + 1) (a^{2} - a - 2) \geq 0 \Rightarrow (a + 1)^{2} (a - 2) \geq 0$
$∴ a \in {- 1} \cup [2, \infty)$

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