Find the roots of the following equation, if they exist, by applying the quadratic formula:
$x^{2} - (\sqrt{3} + 1) x + \sqrt{3} = 0$

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Solution

Roots are given by $\frac{- b \pm \sqrt{b^{2} - 4 a c}}{2 a}$

$= \frac{(\sqrt{3} + 1) \pm \sqrt{(\sqrt{3} + 1)^{2} - 4 \times 1 \times \sqrt{3}}}{2 \times 1}$

$= \frac{(\sqrt{3} + 1) \pm (\sqrt{3} - 1)}{2}$

$= \sqrt{3}, 1$

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