z^2-(5+i)z+8-i=0

解答

z^{2} - (5 + i) z + 8 - i = 0

z = 1.49224 \dots - 1.23653 \dots i, z = 3.50775 \dots + 2.23653 \dots i

求解步骤

z^{2} - (5 + i) z + 8 - i = 0

替代

z = x + y i

(x + y i)^{2} - (5 + i) (x + y i) + 8 - i = 0

展开

(x + y i)^{2} - (5 + i) (x + y i) + 8 - i : (x^{2} - y^{2} - 5 x + y + 8) + i (- 1 - x - 5 y + 2 x y)

(x^{2} - y^{2} - 5 x + y + 8) + i (- 1 - x - 5 y + 2 x y) = 0

将

0

改写成标准复数形式:

0 + 0 i

(x^{2} - y^{2} - 5 x + y + 8) + i (- 1 - x - 5 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} - 5 x + y + 8 = 0 - 1 - x - 5 y + 2 x y = 0]

[x^{2} - y^{2} - 5 x + y + 8 = 0 - 1 - x - 5 y + 2 x y = 0] : (x = 1.49224 \dots, x = 3.50775 \dots, y = - 1.23653 \dots y = 2.23653 \dots)

z = x + y i

代回

z = 1.49224 \dots - 1.23653 \dots i, z = 3.50775 \dots + 2.23653 \dots i

x^{2} + 3 (1 - i) x - 5 i = 0

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

(9 - 8 i) - 6 = x - (2 + y i)

(x + i y) (4 i) = 8

\frac{1285}{8} - 60 i = 1 0^{6} M^{2}