Using the factorisation method, find the roots of the following quadratic equation:
$\frac{x}{x - 1} + \frac{x - 1}{x} = \frac{5}{2}$

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Solution

$\frac{x}{x - 1} + \frac{x - 1}{x} = \frac{5}{2}$ $(x \neq 1, 0)$
$\Rightarrow 2 x^{2} + 2 x^{2} - 4 x + 2 = 5 x^{2} - 5 x$
$\Rightarrow x^{2} - x - 2 = 0$
$\Rightarrow x^{2} - x - 2 (1) = 0$
$\Rightarrow x^{2} - 2 x + x - 2 = 0$
$\Rightarrow x (x - 2) + 1 (x - 2) = 0$
$\Rightarrow (x + 1) (x - 2) = 0$
$\Rightarrow x = - 1, 2$

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