0=10^5(10+ix)(1000+ix)

解答

0 = 1 0^{5} (10 + i x) (1000 + i x)

x = 1000 i, x = 10 i

求解步骤

0 = 1 0^{5} (10 + i x) (1000 + i x)

替代

x = a + bi

0 = 1 0^{5} (10 + i (a + bi)) (1000 + i (a + bi))

展开

1 0^{5} (10 + i (a + bi)) (1000 + i (a + bi)) : (- 100000 a^{2} - 101000000 b + 100000 b^{2} + 1000000000) + i (101000000 a - 200000 ab)

0 = (- 100000 a^{2} - 101000000 b + 100000 b^{2} + 1000000000) + i (101000000 a - 200000 ab)

将

0

改写成标准复数形式:

0 + 0 i

0 + 0 i = (- 100000 a^{2} - 101000000 b + 100000 b^{2} + 1000000000) + i (101000000 a - 200000 ab)

复数仅在实部和虚部均相等时才相等改写为方程组:

[0 = - 100000 a^{2} - 101000000 b + 100000 b^{2} + 1000000000 0 = 101000000 a - 200000 ab]

[0 = - 100000 a^{2} - 101000000 b + 100000 b^{2} + 1000000000 0 = 101000000 a - 200000 ab] : (a = 0, a = 0, b = 1000 b = 10)

x = a + bi

代回

x = 1000 i, x = 10 i

- x^{2} = 2 i

x^{2} = 1184 + 125.6637 i

2 w^{2} + (- 6 + 7 i) w - 11 + 2 i = 0

x^{2} + (2 - 2 i) x - (4 + 2 i) = 0

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