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  4. The real roots of |x|3 - 3x2 + 3 |x| - 2 = 0 are

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Q. The real roots of
$|x|^3 - 3x^2 + 3 |x| - 2 = 0$ are

Complex Numbers and Quadratic Equations

Solution:

$|x|^3 - 3|x^2 + 3 x| - 2$ = 0 [Clearly $|x| = 2 $ satisfies it]
$\Rightarrow \, (|x | -2)(|x | ^2- |x | + 1) = 0 $
$\Rightarrow \, | x | = 2 $ or $| x |^2 - | x | +1 = 0 $
$\Rightarrow \, | x | = 2\,[\because \, | x |^2 - | x | + 1 = 0]$ has no real roots;
Disc = 1 - 4 = -ve]
$\Rightarrow \, x = \pm 2$