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Express the equation $\frac{1}{x} - \frac{1}{x - 2} = 3, (x \neq 0, 2)$ as a quadratic equation in standard form. Hence, find the roots of the quadratic equation so obtained.
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$\frac{x - 2 - x}{x (x - 2)} = 3$ [1 mark]
$\Rightarrow 3x^2 - 6x + 2 = 0$ [1 mark]
$\text{Discriminant} = 36 - 24 = 12$ [1 mark]
$\text{Roots are } \frac{6 + \sqrt{12}}{6} \text{ and } \frac{6 - \sqrt{12}}{6}$
$\text{or } 1 + \frac{\sqrt{3}}{3} \text{ and } 1 - \frac{\sqrt{3}}{3}$ [1+1 marks]
$\Rightarrow 3x^2 - 6x + 2 = 0$ [1 mark]
$\text{Discriminant} = 36 - 24 = 12$ [1 mark]
$\text{Roots are } \frac{6 + \sqrt{12}}{6} \text{ and } \frac{6 - \sqrt{12}}{6}$
$\text{or } 1 + \frac{\sqrt{3}}{3} \text{ and } 1 - \frac{\sqrt{3}}{3}$ [1+1 marks]