What is the answer to f^2= 1/2*(f+i) ?

The answer to f^2= 1/2*(f+i) is f=-0.28216…-0.46978…i,f=0.78216…+0.46978…i

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f^2= 1/2*(f+i)

f^{2} = \frac{1}{2} \cdot (f + i)

f = - 0.28216 \dots - 0.46978 \dots i, f = 0.78216 \dots + 0.46978 \dots i

Solution steps

f^{2} = \frac{1}{2} (f + i)

Substitute

f = x + y i

(x + y i)^{2} = \frac{1}{2} (x + y i + i)

Expand

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

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

[x^{2} - y^{2} = \frac{x}{2} 2 x y = \frac{1 + y}{2}]

[x^{2} - y^{2} = \frac{x}{2} 2 x y = \frac{1 + y}{2}] : (x = - 0.28216 \dots, x = 0.78216 \dots, y = - 0.46978 \dots y = 0.46978 \dots)

Substitute back

f = x + y i

f = - 0.28216 \dots - 0.46978 \dots i, f = 0.78216 \dots + 0.46978 \dots i

(x + 3) + 4 i = 2 + (y - 2) i

2 z^{2} - 7 z + 4 = i (2 z + 1)

x + 2 + 7 i = y i - 2

i (x) = - x^{2} + 80 x + 650

(1 + i) (x + 2 y) + (3 - 2 i) (x - y) = 8 + 3 i 3 i

What is the answer to f^2= 1/2*(f+i) ?
The answer to f^2= 1/2*(f+i) is f=-0.28216…-0.46978…i,f=0.78216…+0.46978…i