Additional Concepts
-$$x ^ { 3 } - ( - 6 \sqrt { 3 } + 6 i ) = 0$$
A) $\sqrt [ 3 ] { 12 } \left( \cos 50 ^ { \circ } + i \sin 50 ^ { \circ } \right) , \sqrt [ 3 ] { 12 } \left( \cos 170 ^ { \circ } + i \sin 170 ^ { \circ } \right) , \sqrt [ 3 ] { 12 } \left( \cos 290 ^ { \circ } + i \sin 290 ^ { \circ } \right)$
B) $\sqrt [ 3 ] { 6 } \left( \cos 70 ^ { \circ } + i \sin 70 ^ { \circ } \right) , \sqrt [ 3 ] { 6 } \left( \cos 190 ^ { \circ } + i \sin 190 ^ { \circ } \right) , \sqrt [ 3 ] { 6 } \left( \cos 310 ^ { \circ } + i \sin 310 ^ { \circ } \right)$
C) $\sqrt [ 3 ] { 12 } \left( \cos 50 ^ { \circ } + i \sin 50 ^ { \circ } \right) , \sqrt [ 3 ] { 12 } \left( \cos 170 ^ { \circ } + i \sin 170 ^ { \circ } \right) , \sqrt [ 3 ] { 12 } \left( \cos 270 ^ { \circ } + i \sin 270 ^ { \circ } \right)$
D) $\sqrt { 6 } \left( \cos 70 ^ { \circ } + i \sin 70 ^ { \circ } \right) , \sqrt { 6 } \left( \cos 190 ^ { \circ } + i \sin 190 ^ { \circ } \right) , \sqrt { 6 } \left( \cos 310 ^ { \circ } + i \sin 310 ^ { \circ } \right)$

Question

Quizplus · Accepted Answer

The equation can be rewritten in polar form as $r(\cos \theta + i\sin \theta) = 12(\cos 150^\circ + i\sin 150^\circ)$, where $r = \sqrt[3]{12}$ and the angles are $150^\circ + 360^\circ k / 3$, with $k = 0, 1, 2$. This gives the roots at $50^\circ, 170^\circ, 290^\circ$.

Additional Concepts - x3−(−63+6i)=0x ^ { 3 } - ( - 6 \sqrt { 3 } + 6 i ) = 0x3−(−63​+6i)=0

Additional Concepts
- $x ^ { 3 } - ( - 6 \sqrt { 3 } + 6 i ) = 0$