Solve $\frac{2}{x^{2}} - \frac{5}{x} + 2 = 0$ by factorisation method. Find the roots of (PS)
$\frac{1}{x} - \frac{1}{x - 2} = 3 x \neq 0, 2$ .

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Solution

Given,

$\frac{2}{x^{2}} - \frac{5}{x} + 2 = 0$

$2 - 5 x + 2 x^{2} = 0$

$2 x^{2} - 5 x + 2 = 0$

$(x - 2) (2 x - 1) = 0$

$x = 2, \frac{1}{2}$

Given,

$\frac{1}{x} - \frac{1}{x - 2} = 3$

$x - 2 - x = 3 (x) (x - 2)$

$- 2 = 3 x^{2} - 6 x$

$3 x^{2} - 6 x + 2 = 0$

$x = \frac{- (- 6) \pm \sqrt{{(- 6)}^{2} - 4 \cdot 3 \cdot 2}}{2 \cdot 3}$

$= \frac{6 \pm \sqrt{12}}{6}$

$= \frac{2 (3 \pm \sqrt{3})}{6}$

$x = \frac{3 \pm \sqrt{3}}{3}$

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Find the roots of the quadratic equation $x^{2} - 3 x + 2 = 0$ , using factorisation method.

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