any one can solve this equation?

Solve 920 = {50[1 - (1 + r)-12]}/r + 1000/(1 + r)12.


920 = {50[1 - (1 + r)-12]}/r + 1000/(1 + r)12

920 = [50 - 50/(1 + r)12]/r + 1000/(1 + r)12

920 = [50(1 + r)12 - 50]/[r(1 + r)12] + 1000/(1 + r)12

920 = [50(1 + r)12 + 1000r - 50]/[r(1 + r)12]

920r(1 + r)12 = 50(1 + r)12 + 1000r - 50

920r(1 + r)12 - 50(1 + r)12 - 1000r + 50 = 0

92r(1 + r)12 - 5(1 + r)12 - 100r + 5 = 0


Let x = 1 + r,

Then 92(x - 1)x12 - 5x12 - 100(x - 1) + 5 = 0

92x13 - 92x12 - 5x12 - 100x + 100 + 5 = 0

92x13 - 97x12 - 100x + 105 = 0



這是一條13次方程，而且無法factorize至(Ax¹² + B)(Cx + D)這樣式，我實在冇把握可以找到所有exact values of roots。

還是交由MATLAB去solve吧！不過只能找到近似值。


圖片參考：http://i212.photobucket.com/albums/cc82/doraemonpaul/yahoo_knowledge/solve_in_matlab.jpg



∴The roots of the equation 92r(1 + r)12 - 5(1 + r)12 - 100r + 5 = 0 are

（答案準確至32位有效數字，次序經本人重新排列，數值按遞降次序排列，complex roots go with pairs）

r = 0
or r ≈ 0.059516906366860088312503329829791
or - 2.0067954815900609154630654004562
or - 0.12781411494890710376209639738778 + 0.50281423543977315369887439106780i
or - 0.12781411494890710376209639738778 - 0.50281423543977315369887439106780i
or - 0.49634840578813211126960146024766 + 0.87175062801067851529952202046920i
or - 0.49634840578813211126960146024766 - 0.87175062801067851529952202046920i
or - 0.99982679276923226260974690682080 + 1.0067916481795949539480475438424i
or - 0.99982679276923226260974690682080 - 1.0067916481795949539480475438424i
or - 1.5033104748448503799769479379725 + 0.87195937078246818016194141881189i
or - 1.5033104748448503799769479379725 - 0.87195937078246818016194141881189i
or - 1.8718870109937994679367610448667 + 0.50343781110739700152766809624716i
or - 1.8718870109937994679367610448667 - 0.50343781110739700152766809624716i

不過，當我們睇返方程920 = {50[1 - (1 + r)-12]}/r + 1000/(1 + r)12嘅時候，

發覺它有分母r和(1 + r)¹²，而大家都知道，分母在任何時候都不可以是0。

換言之，r不可以是0或是- 1。Thus r = 0 should be rejected.

∴The roots of the equation 920 = {50[1 - (1 + r)-12]}/r + 1000/(1 + r)12 are

（答案準確至32位有效數字，次序經本人重新排列，數值按遞降次序排列，complex roots go with pairs）

r ≈ 0.059516906366860088312503329829791
or - 2.0067954815900609154630654004562
or - 0.12781411494890710376209639738778 + 0.50281423543977315369887439106780i
or - 0.12781411494890710376209639738778 - 0.50281423543977315369887439106780i
or - 0.49634840578813211126960146024766 + 0.87175062801067851529952202046920i
or - 0.49634840578813211126960146024766 - 0.87175062801067851529952202046920i
or - 0.99982679276923226260974690682080 + 1.0067916481795949539480475438424i
or - 0.99982679276923226260974690682080 - 1.0067916481795949539480475438424i
or - 1.5033104748448503799769479379725 + 0.87195937078246818016194141881189i
or - 1.5033104748448503799769479379725 - 0.87195937078246818016194141881189i
or - 1.8718870109937994679367610448667 + 0.50343781110739700152766809624716i
or - 1.8718870109937994679367610448667 - 0.50343781110739700152766809624716i