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

解答

∣ 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

求解步骤

∣ z ∣^{2} + i + i 2 i = 6 z

替代

z = x + y i

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

乘开

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

复数仅在实部和虚部均相等时才相等改写为方程组:

[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})

z = x + y i

代回

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

(x + 4 i) (7 + y i) = 23 + 43 i