If ( 3 + i ) 100 = 2 99 ( p + i q ) , then p and q are roots of the equation :

$(\sqrt{3}+i{)}^{100}=2^{99}(p+iq)$

$2^{100}{\left(\cos \frac{\mathrm{\pi }}{6}+i\sin \frac{\mathrm{\pi }}{6}\right)}^{100}=2^{99}\left(p+iq\right)$

$2^{100}{e}^{i\frac{50\pi }{3}}=2^{99}\left(p+iq\right)$

$2\left(\cos \frac{50\mathrm{\pi }}{3}+i\sin \frac{50\mathrm{\pi }}{3}\right)=p\mathit{+}iq$

$2\left(\cos \left(17\pi -\frac{\mathrm{\pi }}{3}\right)+i\sin \left(17\pi -\frac{\mathrm{\pi }}{3}\right)\right)=p\mathit{+}iq$

$2\left(-\cos \frac{\mathrm{\pi }}{3}+i\sin \frac{\mathrm{\pi }}{3}\right)=p\mathit{+}iq$

$2\left(\frac{-1}{2}+\frac{i\sqrt{3}}{2}\right)=p+iq$

$p=-1 \& q=\sqrt{3}$

$p$ and $q$ Roots of the equation
$x^2-(\sqrt{3}-1)x-\sqrt{3}=0$