Let $\mathrm{p,q\in\mathbb{R}}$ and ${\left( {1 - \sqrt 3 i} \right)^{200}} = {2^{199}}(p + iq),i = \sqrt { - 1}$ then $\mathrm{p+q+q^2}$ and $\mathrm{p-q+q^2}$ are roots of the equation.

<p>${\left( {1 - \sqrt 3 i} \right)^{200}}$</p> <p>$= {\left[ {2\left( {{1 \over 2} - {{\sqrt 3 } \over 2}i} \right)} \right]^{200}}$</p> <p>$= {2^{200}}{\left( {\cos {\pi \over 3} - i\sin {\pi \over 3}} \right)^{200}}$</p> <p>$= {2^{200}}\left( {\cos {{200\pi } \over 3} - i\sin {{200\pi } \over 3}} \right)$</p> <p>$$ = {2^{200}}\left( {\cos \left( {66\pi + {{2\pi } \over 3}} \right) - i\sin \left( {66\theta + {{2\pi } \over 3}} \right)} \right)$$</p> <p>$= {2^{200}}\left( {\cos {{2\pi } \over 3} - i\sin {{2\pi } \over 3}} \right)$</p> <p>$= {2^{200}}\left( { - {1 \over 2} - {{\sqrt 3 } \over 2}i} \right)$</p> <p>$= {2^{199}}\left( { - 1 - \sqrt 3 i} \right)$</p> <p>$= {2^{199}}\left( {p + iq} \right)$</p> <p>$\therefore$ p = $-$1 and q = $- \sqrt3$</p> <p>Now, $p - q + {q^2} = - 1 + \sqrt 3 + 3 = 2 + \sqrt 3 = \alpha$</p> <p>and $p + q + {q^2} = - 1 - \sqrt 3 + 3 = 2 - \sqrt 3 = \beta$</p> <p>$\therefore$ $\alpha + \beta = 4$</p> <p>$$\alpha \beta = \left( {2 + \sqrt 3 } \right)\left( {2 - \sqrt 3 } \right) = 4 - 3 = 1$$</p> <p>$\therefore$ Quadratic equation is</p> <p>${x^2} - (\alpha + \beta )x + \alpha \beta = 0$</p> <p>$\Rightarrow {x^2} - 4x + 1 = 0$</p>