Solve the equation. $x ^ { 2 } - i = 0$
A) $- \frac { \sqrt { 2 } } { 2 } - \frac { \sqrt { 2 } } { 2 } i , \frac { \sqrt { 2 } } { 2 } + \frac { \sqrt { 2 } } { 2 } i$
B) $- \sqrt { 2 } - 2 i , \sqrt { 2 } + 2 i$
C) $0 - \frac { \sqrt { 2 } } { 2 } i , 0 + \frac { \sqrt { 2 } } { 2 } i$
D) $- 1 - 2 i , 1 + 2 i$
E) $- i , i$

Question

Quizplus · Accepted Answer

We can solve this by completing the square:
\begin{align*}
x^2-i&=0\
x^2&=i\
x^2+i\left(\frac{1}{2}ight)^2&=i\left(\frac{1}{2}ight)^2+\left(\frac{1}{2}ight)^2\
\left(x+\frac{1}{2}iight)^2&=\frac{1}{4}(1+i)\
x+\frac{1}{2}i&=\pm\sqrt{\frac{1}{4}(1+i)}=\pm\frac{1}{2}\sqrt{1+i}\
x&=-\frac{1}{2}i\pm\frac{1}{2}\sqrt{2}\left(\frac{1}{\sqrt{2}}+\frac{1}{\sqrt{2}}iight)\
&=-\frac{1}{2}\left(1+\sqrt{2}ight)\pm\frac{1}{2}\sqrt{2}\left(1+iight)\
\end{align*}
Therefore, the solutions are:
$$x=-\frac{1}{2}\left(1+\sqrt{2}ight)-\frac{1}{2}\sqrt{2}i,\hspace{1cm}x=-\frac{1}{2}\left(1+\sqrt{2}ight)+\frac{1}{2}\sqrt{2}i$$
Which matches answer choice A.

Solve the Equation x2−i=0x ^ { 2 } - i = 0x2−i=0

Solve the Equation $x ^ { 2 } - i = 0$