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受欢迎的代数 >

z^6-(3-2i)z^3+2-2i=0

解答

z^{6} - (3 - 2 i) z^{3} + 2 - 2 i = 0

解答

z = 1, z = - \frac{1}{2} + \frac{3}{2} i, z = - \frac{1}{2} - \frac{3}{2} i, z = \frac{1 + 3}{2} + i \frac{1 - 3}{2}, z = \frac{1 - 3}{2} + i \frac{1 + 3}{2}, z = - 1 - i

求解步骤

z^{6} - (3 - 2 i) z^{3} + 2 - 2 i = 0

展开

z^{6} - (3 - 2 i) z^{3} + 2 - 2 i : (z^{6} - 3 z^{3} + 2) + i (- 2 + 2 z^{3})

(z^{6} - 3 z^{3} + 2) + i (- 2 + 2 z^{3}) = 0

改写成标准形式

a_{n} x^{n} + \dots + a_{1} x + a_{0} = 0

z^{6} + (- 3 + 2 i) z^{3} + 2 - 2 i = 0

用

u = z^{3}

和

u^{2} = z^{6}

改写方程式

u^{2} + (- 3 + 2 i) u + 2 - 2 i = 0

解

u^{2} + (- 3 + 2 i) u + 2 - 2 i = 0 : u = 1, u = 2 - 2 i

u = 1, u = 2 - 2 i

代回

u = z^{3}

，求解

z

解

z^{3} = 1 : z = 1, z = - \frac{1}{2} + \frac{3}{2} i, z = - \frac{1}{2} - \frac{3}{2} i

解

z^{3} = 2 - 2 i : z = \frac{1 + 3}{2} + i \frac{1 - 3}{2}, z = \frac{1 - 3}{2} + i \frac{1 + 3}{2}, z = - 1 - i

解为

z = 1, z = - \frac{1}{2} + \frac{3}{2} i, z = - \frac{1}{2} - \frac{3}{2} i, z = \frac{1 + 3}{2} + i \frac{1 - 3}{2}, z = \frac{1 - 3}{2} + i \frac{1 + 3}{2}, z = - 1 - i

流行的例子

4(x^2-x)^2-11(x^2-x)+6=0

4 (x^{2} - x)^{2} - 11 (x^{2} - x) + 6 = 0

5 y^{3} + 33 y^{2} = 14 y

20 x^{3} - 15 x = 3

(x^2-x-3)^2-2(x^2-x)+3=0

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

10 = x^{10}