z^2+z+3+15i=0

解答

z^{2} + z + 3 + 15 i = 0

z = 2 - 3 i, z = - 3 + 3 i

求解步骤

z^{2} + z + 3 + 15 i = 0

替代

z = x + y i

(x + y i)^{2} + x + y i + 3 + 15 i = 0

展开

(x + y i)^{2} + x + y i + 3 + 15 i : (x^{2} - y^{2} + x + 3) + i (15 + y + 2 x y)

(x^{2} - y^{2} + x + 3) + i (15 + y + 2 x y) = 0

将

0

改写成标准复数形式:

0 + 0 i

(x^{2} - y^{2} + x + 3) + i (15 + y + 2 x y) = 0 + 0 i

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} + x + 3 = 0 15 + y + 2 x y = 0]

[x^{2} - y^{2} + x + 3 = 0 15 + y + 2 x y = 0] : (x = 2, x = - 3, y = - 3 y = 3)

z = x + y i

代回

z = 2 - 3 i, z = - 3 + 3 i

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

7 + 3 x - 3 i = 8 x - 6 y i - 8 + 9 i

x + y i - \frac{1}{x + y i} = 5 + 5 i

\frac{x + i y}{x - i y} = - i

(a + i) (b - 3 i) = 7 - 11 i