What is the answer to x^2=-i ?

The answer to x^2=-i is x= 1/(sqrt(2))-1/(sqrt(2))i,x=-1/(sqrt(2))+1/(sqrt(2))i

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x^2=-i

x^{2} = - i

x = \frac{1}{2} - \frac{1}{2} i, x = - \frac{1}{2} + \frac{1}{2} i

Solution steps

x^{2} = - i

Substitute

x = a + bi

(a + bi)^{2} = - i

Expand

(a + bi)^{2} : (a^{2} - b^{2}) + 2 iab

(a^{2} - b^{2}) + 2 iab = - i

Rewrite

- i

in standard complex form:

0 - i

(a^{2} - b^{2}) + 2 iab = 0 - i

Complex numbers can be equal only if their real and imaginary parts are equalRewrite as system of equations:

[a^{2} - b^{2} = 0 2 ab = - 1]

[a^{2} - b^{2} = 0 2 ab = - 1] : (a = \frac{1}{2}, a = - \frac{1}{2}, b = - \frac{1}{2} b = \frac{1}{2})

Substitute back

x = a + bi

x = \frac{1}{2} - \frac{1}{2} i, x = - \frac{1}{2} + \frac{1}{2} i

z^{2} + 2 z + 1 + i = 0

z^{- 1} = \frac{1}{3 + 2 i}

z^{2} - 2 i z - 3 = 0

x + i y = 0

(x + 2) + (2 y - 1) i = - 1 + 5 i

What is the answer to x^2=-i ?
The answer to x^2=-i is x= 1/(sqrt(2))-1/(sqrt(2))i,x=-1/(sqrt(2))+1/(sqrt(2))i