數學的方程式

1. 3x^3+x=1
Sol
3x^3+x=1
3x^3+x－1=0
(3x^3－1.6097643x^2)+(1.6097643x^2－0.863769x)+(1.863769x－1)=0
3x^2(x－0.5365811)+1.6097643x(x－0.5365811)+1.863769(x－0.5365811)=0
(x－0.5365811)(3x^2+1.6097643x+1.863769)=0
x=0.5365811 or x=(－1.6097643+/-√(1.6097643^2-4*3*1.863769))/6
x=0.5365811 or x=(－1.6097643+/-√－19.773887)/6
x=0.5365811or x=－05365811+/-0.74113i

2.3x^3－1=0
Sol
3x^3－1=0
x^3－(1/3)=0
x^3－(0.69336)^3=0
(x－0.69336)(x^2+0.69336x+0.480748)=0
x=0.69336 or x=(－0.69336+/-√0.69336^2－4*1*0.480748))/2
x=00.69336 or x=－0.34668+/-√－1.442244
x=00.69336or x=－0.34668+/-1.209346i

Question 1: 3x^3+x =1 This is a cubic equation.
ax^3+bx^2+cx+d = 0
3x^3 + x -1 = 0 where a = 3, b = 0 c = 1, d = -1
By Descarte’s rule, 有一個real root 和兩個unreal roots.

方法1: Cardano 公式
x = {q + [q^2 +(r-p^2)^3]^1/2}^1/3 + {q - [q^2 + (r-p^2)^3]^1/2}^1/3 + pwhere p = -b/(3a), q = p^3+ (bc-3ad)/(6a^2), r = c/(3a)p = 0, q = 0 + (0 -3(3)(-1))/(6)(9)= 9/54 = 1/6, r = 1/9x = {q + [q^2 +(r-p^2)^3]^1/2}^1/3 + {q - [q^2 + (r-p^2)^3]^1/2}^1/3 + px = {1/6 + [1/36 + (1/9)^3]^1/2}^1/3 + {1/6 - [1/36 + (1/9)^3]^1/2}^1/3 x = {1/6 + [1/36 + 1/729)]^1/2}^1/3 + {1/6 - [1/36 + 1/729]^1/2}^1/3 x = {1/6 + [85/2916]^1/2}^1/3 + {1/6 - [85/2916]^1/2}^1/3 x = 0.696168846 + -0.159603667
x = 0.53656565178

The two unreal roots are -0.2682826 +-0.7411207i

Check: x = 0.53656565178, 3x^3 + x -1 = 0
3(0.53656565178^3) + 0.53656565178 -1 = 0.00000174646

方法2: Newton’s method (微積分)
x(n+1) = xn - f(x) / f '(x) where f '(x) is the derivative.
xn = x - (3x^3 + x - 1)/(9x^2 + 1),
Trial 1: put x = 0.54, xn = 0.536580951
Trial 2: put x = .536580951, xn = 0.536565165 (even more accurate)

Check: x = 0.536565165, 3x^3 + x -1 = 0,
3(0.536565165^3) + 0.536565165 -1 = 0.00000000117

Question 2: 3x^3-1= 0
x^3 = 1/3
x =1/[(3)^(1/3)] = 0.693361274
x = 0.693361274 (one real root)

x^3 – 1/3 = 0
x^3 – (0.693361274)^3 = 0 (difference of cubes)
(x - 0.693361274)(x^2 +0.693361274x + 0.693361274^2)
(x^2 +0.693361274x + 0.693361274^2) Using quadratic formula, solving for two unreal roots [a = 1, b = 0.693361274, c = 0.480749856]

The other two unreal roots are -0.346680637 +- 0.600468477i

You can also solve cubic equation by Vieta’s substitution (請查閱工程數學教科書)
or write a computer program using Fortran language which has a built-in complex number function.


2012-01-15 11:51:38 補充：
To 回答者：螞蟻雄兵. No offence.
For question 2 in the unreal roots, x=－0.34668+/-1.209346i
you forget to divide the imaginary part by 2. I sometimes make similar mistake.
(1.209346/2) i

2012-01-15 21:29:32 補充：
補充:
第二問題可用Cardano 公式計算
a = 3, b =0, c = 0, d = -1
p = 0, q = 1/6, r = 0
x = {q + [q^2 +(r-p^2)^3]^1/2}^1/3 + {q - [q^2 + (r-p^2)^3]^1/2}^1/3 + p
x = {1/6 + 1/6}^1/3 + {1/6 – 1/6}^1/3
x = {1/3}^1/3
x = 0.693361274