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受欢迎的代数 >

x^6+4x^3+16=0

解答

x^{6} + 4 x^{3} + 16 = 0

解答

x = 34 cos (\frac{2 π}{9}) + 34 i sin (\frac{2 π}{9}), x = 34 cos (\frac{8 π}{9}) + 34 i sin (\frac{8 π}{9}), x = 34 cos (\frac{14 π}{9}) + 34 i sin (\frac{14 π}{9}), x = 34 cos (\frac{2 π}{9}) - 34 i sin (\frac{2 π}{9}), x = 34 cos (\frac{4 π}{9}) + 34 i sin (\frac{4 π}{9}), x = 34 cos (\frac{10 π}{9}) + 34 i sin (\frac{10 π}{9})

求解步骤

x^{6} + 4 x^{3} + 16 = 0

用

u = x^{3}

和

u^{2} = x^{6}

改写方程式

u^{2} + 4 u + 16 = 0

解

u^{2} + 4 u + 16 = 0 : u = - 2 + 23 i, u = - 2 - 23 i

u = - 2 + 23 i, u = - 2 - 23 i

代回

u = x^{3}

，求解

x

解

x^{3} = - 2 + 23 i : x = 34 cos (\frac{2 π}{9}) + 34 i sin (\frac{2 π}{9}), x = 34 cos (\frac{8 π}{9}) + 34 i sin (\frac{8 π}{9}), x = 34 cos (\frac{14 π}{9}) + 34 i sin (\frac{14 π}{9})

解

x^{3} = - 2 - 23 i : x = 34 cos (\frac{2 π}{9}) - 34 i sin (\frac{2 π}{9}), x = 34 cos (\frac{4 π}{9}) + 34 i sin (\frac{4 π}{9}), x = 34 cos (\frac{10 π}{9}) + 34 i sin (\frac{10 π}{9})

解为

x = 34 cos (\frac{2 π}{9}) + 34 i sin (\frac{2 π}{9}), x = 34 cos (\frac{8 π}{9}) + 34 i sin (\frac{8 π}{9}), x = 34 cos (\frac{14 π}{9}) + 34 i sin (\frac{14 π}{9}), x = 34 cos (\frac{2 π}{9}) - 34 i sin (\frac{2 π}{9}), x = 34 cos (\frac{4 π}{9}) + 34 i sin (\frac{4 π}{9}), x = 34 cos (\frac{10 π}{9}) + 34 i sin (\frac{10 π}{9})

作图

流行的例子

(x-7)(x^2-36)=0

(x - 7) (x^{2} - 36) = 0

x^3-x^2-22x+40=0

x^{3} - x^{2} - 22 x + 40 = 0

k^{3} + 4 = - 16

x^{3} - x^{2} - 5 x = - 5

x^{3} - 1 = x - 1