已知实数a、b、c、d互不相等,且a+ 1/b=b+ 1/c=c+ 1/d=d+ 1/a.求abc已知实数a、b、c、d互不相等,且a+ 1/b=b+ 1/c=c+ 1/d=d+ 1/a.求abcd的值.
来源：学生作业帮助网 编辑：六六作业网 时间：2025/02/01 21:54:36
已知实数a、b、c、d互不相等,且a+ 1/b=b+ 1/c=c+ 1/d=d+ 1/a.求abc已知实数a、b、c、d互不相等,且a+ 1/b=b+ 1/c=c+ 1/d=d+ 1/a.求abcd的值.
已知实数a、b、c、d互不相等,且a+ 1/b=b+ 1/c=c+ 1/d=d+ 1/a.求abc
已知实数a、b、c、d互不相等,且a+ 1/b=b+ 1/c=c+ 1/d=d+ 1/a.求abcd的值.
已知实数a、b、c、d互不相等,且a+ 1/b=b+ 1/c=c+ 1/d=d+ 1/a.求abc已知实数a、b、c、d互不相等,且a+ 1/b=b+ 1/c=c+ 1/d=d+ 1/a.求abcd的值.
实数a、b、c、d互不相等,且a+ 1/b=b+ 1/c=c+ 1/d=d+ 1/a,设为t,则
a=t-1/b=(bt-1)/b,
d=t-1/a=t-b/(bt-1)=(bt^2-t-b)/(bt-1),
c=t-1/d=t-(bt-1)/(bt^2-t-b)=(bt^3-t^2-2bt+1)/(bt^2-t-b),
b=t-1/c=t-(bt^2-t-b)/(bt^3-t^2-2bt+1)=(bt^4-t^3-3bt^2+2t+b)/(bt^3-t^2-2bt+1),
∴bt^4-t^3-3bt^2+2t+b=b^2*t^3-bt^2-2b^2*t+b,
∴bt^4-(b^2+1)t^3-2bt^2+2(b^2+1)t=0,
∴t(bt-b^2-1)(t^2-2)=0,
解得t1=0,t2=b+1/b,t3=√2,t4=-√2.
1)t1=0时a+ 1/b=b+ 1/c=0,
a=-1/b,b=-1/c,
∴a=c,舍
2)t2=b+1/b时a+ 1/b=b+1/b,a=b,舍.
3)t3=√2时a+ 1/b=b+ 1/c=c+ 1/d=d+ 1/a=√2,
d=√2-1/a=(a√2-1)/a,
c=√2-1/d=√2-a/(a√2-1)=(a-√2)/(a√2-1),
b=√2-1/c=√2-(a√2-1)/(a-√2)=1/(√2-a),
∴abcd=-1,
4)t4=-√2时a+ 1/b=b+ 1/c=c+ 1/d=d+ 1/a=-√2,
剩下部分留给您练习.