What is the answer to |z|^2+i+i2i=6z ?

The answer to |z|^2+i+i2i=6z is z=(18+sqrt(395))/6+1/6 i,z=(18-sqrt(395))/6+1/6 i

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|z|^2+i+i2i=6z

∣ z ∣^{2} + i + i 2 i = 6 z

z = \frac{18 + 395}{6} + \frac{1}{6} i, z = \frac{18 - 395}{6} + \frac{1}{6} i

Solution steps

∣ z ∣^{2} + i + i 2 i = 6 z

Substitute

z = x + y i

∣ (x + y i) ∣^{2} + i + i 2 i = 6 (x + y i)

Expand

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

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

[x^{2} + y^{2} - 2 = 6 x 1 = 6 y]

[x^{2} + y^{2} - 2 = 6 x 1 = 6 y] : (x = \frac{18 + 395}{6}, x = \frac{18 - 395}{6}, y = \frac{1}{6} y = \frac{1}{6})

Substitute back

z = x + y i

z = \frac{18 + 395}{6} + \frac{1}{6} i, z = \frac{18 - 395}{6} + \frac{1}{6} i

\frac{( 1 + 3 i )}{z + 5} = i

x^{2} - 9 i = 0

(x + 2 - 1) (x - 2 i) = 0

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

(p + 3 i) \cdot (4 + q i) = 15 + 9 i

What is the answer to |z|^2+i+i2i=6z ?
The answer to |z|^2+i+i2i=6z is z=(18+sqrt(395))/6+1/6 i,z=(18-sqrt(395))/6+1/6 i