Find all solutions of the equation. Leave answers in trigonometric form.
-$$x ^ { 2 } + 1 - \sqrt { 3 } i = 0$$
A) $\left\{ 2 \right.$ cis $45 ^ { \circ } , 2$ cis $\left. 225 ^ { \circ } \right\}$
B) $\left\{ 2 \right.$ cis $90 ^ { \circ } , 2$ cis $\left. 270 ^ { \circ } \right\}$
C) $\left\{ \sqrt { 2 } \operatorname { cis } 60 ^ { \circ } , \sqrt { 2 } \operatorname { cis } 240 ^ { \circ } \right\}$
D) $\left\{ 2 \sqrt { 2 } \operatorname { cis } 90 ^ { \circ } , - 2 \sqrt { 2 } \operatorname { cis } 270 ^ { \circ } \right\}$

Question

Quizplus · Accepted Answer

First, rewrite the equation as $x^2 = -1 + \sqrt{3}i$. This can be expressed in polar form as $2\text{ cis }60^\circ$ because $2\cos(60^\circ) = 1$ and $2\sin(60^\circ) = \sqrt{3}$, matching the real and imaginary parts respectively. Using De Moivre's theorem, the square roots are $\sqrt{2}\text{ cis }(30^\circ + k\cdot360^\circ/2)$ for $k=0,1$, giving $60^\circ$ and $240^\circ$.

Find All Solutions of the Equation x2+1−3i=0x ^ { 2 } + 1 - \sqrt { 3 } i = 0x2+1−3​i=0

Find All Solutions of the Equation $x ^ { 2 } + 1 - \sqrt { 3 } i = 0$