Solve the following quadratic equation by factorization method :
$x = \frac{1}{2 - \frac{1}{2 - \frac{1}{2 - x}}}$

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Solution

$x = \frac{1}{2 - \frac{1}{2 - \frac{1}{2 - x}}}$
$⟹ x = \frac{1}{2 - \frac{1}{(\frac{4 - 2 x - 1}{2 x})}}$
$⟹ x = \frac{4 - 2 x - 1}{8 - 4 x - 2 - 2 + x}$
$⟹ x = \frac{4 - 2 x - 1}{4 - 3 x}$
$⟹ 4 x - 3 x^{2} = 4 - 2 x - 1$
$⟹ 3 x^{2} - 6 x + 3 = 0$
$⟹ x^{2} - 3 + 1 = 0$
$⟹ (x - 1)^{2} = 0$
$⟹ x = - 1, - 1$

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