Solve the following equations:
$x^{4} - 3 x^{2} - 6 x - 2 = 0$ .

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Solution

Given equation, $x^{4} - 3 x^{2} - 6 x - 2 = 0$

$\Rightarrow x^{3} + 2 x^{3} - 2 x^{3} + 2 x^{2} - 4 x^{2} - x^{2} - 4 x - 2 x - 2 = 0$

$\Rightarrow x^{3} + 2 x^{3} + 2 x^{2} - 2 x^{3} - 4 x^{2} - 4 x - x^{2} - 2 x - 2 = 0 [Rearranging the coefficients]$

$\Rightarrow (x^{2} - 2 x - 1) (x^{2} + 2 x + 2) = 0 [Taking common terms together]$

Solving both the quadratic equations, we have
$x = - 1 \pm \sqrt{2}, - 1 \pm i$

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