Question

解方程1/(x*x+3x+2)+1/(x*x+5x+6)+1/(x*x+7x+12)=1/3x

Answer 1

(1) 1/(x*x+3x+2)+1/(x*x+5x+6)+1/(x*x+7x+12)=1/3x
1/[(x+1)(x+2)]+1/[(x+2)(x+3)]+1/[(x+3)(x+4)]=1/3x
[(x+3)(x+4)+(x+1)(x+4)+(x+1)(x+2)]=1/3x
x²+7x+12+x²+5x+4+x²+3x+2=1/3x
3x²+15x+18=1/3x
9x²+45x+54=x
9x²+44x+54=0
x²+44x=6
x²+44x+22²=484+6
(x+22)²=490
x+22=±7√10
x=±7√10-22
(2)
1/(x*x+3x+2)+1/(x*x+5x+6)+1/(x*x+7x+12)=1/(3x)
1/[(x+1)(x+2)]+1/[(x+2)(x+3)]+1/[(x+3)(x+4)]=1/(3x)
[(x+3)(x+4)+(x+1)(x+4)+(x+1)(x+2)]=1/(3x)
x²+7x+12+x²+5x+4+x²+3x+2=1/(3x)
3x²+15x+18=1/(3x)
9x³+45x²+54x=1
只有一个实数解
x1=0.018240251......

Answer 2

1/(x*x+3x+2)+1/(x*x+5x+6)+1/(x*x+7x+12)=1/3x
1/[(x+2)(x+1)]+1/[(x+2)(x+3)]+1/[(x+3)(x+4)]=1/3x
（x+3)(x+4)+(x+1)(x+4)+(x+1)(x+2)=(x+1)(x+2)(x+3)(x+4)/3x
x^2+7x+12+x^2+5x+4+x^2+3x+2=(x+1)(x+2)(x+3)(x+4)/3x
3x^2+15x+18=(x+1)(x+2)(x+3)(x+4)/3x
x^2+5x+6=(x+1)(x+2)(x+3)(x+4)/9x
(x+2)(x+3)=(x+1)(x+2)(x+3)(x+4)/9x
(x+1)(x+4)=9x
x^2+5x+4=9x
x^2-4x+4=0
(x-2)^2=0
x-2=0
x=2

Answer 3

原方程可化为：
1/[(x+2)(x+1)] + 1/[(x+3)(x+2)] + 1/[(x+4)(x+3)] =1/(3x)
即：1/(x+1) - 1/(x+2) + 1/(x+2) - 1/(x+3) + 1/(x+3) - 1/(x+4)=1/(3x)
1/(x+1) - 1/(x+4)=1/(3x)
(x+4 - x-1)/[(x+1)(x+4)]=1/(3x)
3/[(x+1)(x+4)]=1/(3x)
9x=(x+1)(x+4)
9x=x²+5x+4
x²-4x+4=0
(x-2)²=0
解得：x=2
经检验，原方程的解为：x=2

Answer 4

提取公因式强解。