Find for what value of $a$ equation $x^{2} + 2 x - (a^{3} - 3 a - 3) = 0$ has real roots.

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Solution

Given,

$x^{2} + 2 x - (a^{3} - 3 a - 3) = 0$

$a = 1, b = 2, c = - (a^{3} - 3 a - 3)$

For the roots to be real, condition is,

$b^{2} - 4 a c = 0 \Rightarrow b^{2} = 4 a c$

$∴ 2^{2} = 4 (1) (- (a^{3} - 3 a - 3))$

$4 = - 4 (a^{3} - 3 a - 3)$

$- 1 = a^{3} - 3 a - 3$

$a^{3} - 3 a - 2 = 0$

${(a + 1)}^{2} (a - 2) = 0$

$a = - 1, a = 2$

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