What is the answer to z^2=i(1-0.5^2) ?

The answer to z^2=i(1-0.5^2) is z=(sqrt(3))/(2sqrt(2))+0.61237…i,z=-(sqrt(3))/(2sqrt(2))-0.61237…i

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z^2=i(1-0.5^2)

z^{2} = i (1 - 0. 5^{2})

z = \frac{3}{2 2} + 0.61237 \dots i, z = - \frac{3}{2 2} - 0.61237 \dots i

Solution steps

z^{2} = i (1 - 0. 5^{2})

Substitute

z = x + y i

(x + y i)^{2} = i (1 - 0. 5^{2})

Expand

(x^{2} - y^{2}) + 2 i x y = 0.75 i

Rewrite

0.75 i

in standard complex form:

0 + 0.75 i

(x^{2} - y^{2}) + 2 i x y = 0 + 0.75 i

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

[x^{2} - y^{2} = 0 2 x y = 0.75]

[x^{2} - y^{2} = 0 2 x y = 0.75] : (x = \frac{3}{2 2}, x = - \frac{3}{2 2}, y = 0.61237 \dots y = - 0.61237 \dots)

Substitute back

z = x + y i

z = \frac{3}{2 2} + 0.61237 \dots i, z = - \frac{3}{2 2} - 0.61237 \dots i

x (1 + i) + y (1 - i) = 0

7 i - 3 j = a (i - j) + b (i + j)

x^{2} + 6 x i - 9 = (5 + 12 i) y

3 x + 8 i = 12 - 12

x i + y^{2} (1 + i) = 3 - 2 i

What is the answer to z^2=i(1-0.5^2) ?
The answer to z^2=i(1-0.5^2) is z=(sqrt(3))/(2sqrt(2))+0.61237…i,z=-(sqrt(3))/(2sqrt(2))-0.61237…i