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On line Solution of Monovariate Equation：
Input any unary equation directly, and then click the "Next" button to obtain the solution of the equation.
It supports equations that contain mathematical functions.
Input any unary equation directly, and then click the "Next" button to obtain the solution of the equation.
It supports equations that contain mathematical functions.
Current location：Equations > Monovariate Equation > The history of univariate equation calculation > Answer
Overview: 1 questions will be solved this time.Among them
☆1 equations
[ 1/1 Equation]
Work: Find the solution of equation 4 = (x/(x+1))+((x+1)/(x+2))+((x+2)/(x+3))+((x+3)/(x+4)) .
Question type: Equation
Solution:Original question:
| 4 | = | ( | x | ÷ | ( | x | + | 1 | ) | ) | + | ( | ( | x | + | 1 | ) | ÷ | ( | x | + | 2 | ) | ) | + | ( | ( | x | + | 2 | ) | ÷ | ( | x | + | 3 | ) | ) | + | ( | ( | x | + | 3 | ) | ÷ | ( | x | + | 4 | ) | ) |
| 4 | = | x | ÷ | ( | x | + | 1 | ) | + | ( | ( | x | + | 1 | ) | ÷ | ( | x | + | 2 | ) | ) | + | ( | ( | x | + | 2 | ) | ÷ | ( | x | + | 3 | ) | ) | + | ( | ( | x | + | 3 | ) | ÷ | ( | x | + | 4 | ) | ) |
| Multiply both sides of the equation by: | ( | x | + | 1 | ) |
| 4 | ( | x | + | 1 | ) | = | x | + | ( | ( | x | + | 1 | ) | ÷ | ( | x | + | 2 | ) | ) | ( | x | + | 1 | ) | + | ( | ( | x | + | 2 | ) | ÷ | ( | x | + | 3 | ) | ) | ( | x | + | 1 | ) | + | ( | ( | x | + | 3 | ) | ÷ | ( | x | + | 4 | ) | ) | ( | x | + | 1 | ) |
| 4 | x | + | 4 | × | 1 | = | x | + | ( | ( | x | + | 1 | ) | ÷ | ( | x | + | 2 | ) | ) | ( | x | + | 1 | ) | + | ( | ( | x | + | 2 | ) | ÷ | ( | x | + | 3 | ) | ) | ( | x | + | 1 | ) | + | ( | ( | x | + | 3 | ) | ÷ | ( | x | + | 4 | ) | ) | ( | x | + | 1 | ) |
| 4 | x | + | 4 | × | 1 | = | x | + | ( | x | + | 1 | ) | ÷ | ( | x | + | 2 | ) | × | ( | x | + | 1 | ) | + | ( | ( | x | + | 2 | ) | ÷ | ( | x | + | 3 | ) | ) | ( | x | + | 1 | ) | + | ( | ( | x | + | 3 | ) | ÷ | ( | x | + | 4 | ) | ) | ( | x | + | 1 | ) |
| 4 | x | + | 4 | = | x | + | ( | x | + | 1 | ) | ÷ | ( | x | + | 2 | ) | × | ( | x | + | 1 | ) | + | ( | ( | x | + | 2 | ) | ÷ | ( | x | + | 3 | ) | ) | ( | x | + | 1 | ) | + | ( | ( | x | + | 3 | ) | ÷ | ( | x | + | 4 | ) | ) | ( | x | + | 1 | ) |
| Multiply both sides of the equation by: | ( | x | + | 2 | ) |
| 4 | x | ( | x | + | 2 | ) | + | 4 | ( | x | + | 2 | ) | = | x | ( | x | + | 2 | ) | + | ( | x | + | 1 | ) | ( | x | + | 1 | ) | + | ( | ( | x | + | 2 | ) | ÷ | ( | x | + | 3 | ) | ) | ( | x | + | 1 | ) | ( | x | + | 2 | ) | + | ( | ( | x | + | 3 | ) | ÷ | ( | x | + | 4 | ) | ) | ( | x | + | 1 | ) | ( | x | + | 2 | ) |
| 4 | x | x | + | 4 | x | × | 2 | + | 4 | ( | x | + | 2 | ) | = | x | ( | x | + | 2 | ) | + | ( | x | + | 1 | ) | ( | x | + | 1 | ) | + | ( | ( | x | + | 2 | ) | ÷ | ( | x | + | 3 | ) | ) | ( | x | + | 1 | ) | ( | x | + | 2 | ) | + | ( | ( | x | + | 3 | ) | ÷ | ( | x | + | 4 | ) | ) | ( | x | + | 1 | ) | ( | x | + | 2 | ) |
| 4 | x | x | + | 4 | x | × | 2 | + | 4 | ( | x | + | 2 | ) | = | x | x | + | x | × | 2 | + | ( | x | + | 1 | ) | ( | x | + | 1 | ) | + | ( | ( | x | + | 2 | ) | ÷ | ( | x | + | 3 | ) | ) | ( | x | + | 1 | ) | ( | x | + | 2 | ) | + | ( | ( | x | + | 3 | ) | ÷ | ( | x | + | 4 | ) | ) | ( | x | + | 1 | ) | ( | x | + | 2 | ) |
| 4 | x | x | + | 8 | x | + | 4 | ( | x | + | 2 | ) | = | x | x | + | x | × | 2 | + | ( | x | + | 1 | ) | ( | x | + | 1 | ) | + | ( | ( | x | + | 2 | ) | ÷ | ( | x | + | 3 | ) | ) | ( | x | + | 1 | ) | ( | x | + | 2 | ) | + | ( | ( | x | + | 3 | ) | ÷ | ( | x | + | 4 | ) | ) | ( | x | + | 1 | ) | ( | x | + | 2 | ) |
| 4 | x | x | + | 8 | x | + | 4 | x | + | 4 | × | 2 | = | x | x | + | 2 | x | + | ( | x | + | 1 | ) | ( | x | + | 1 | ) | + | ( | ( | x | + | 2 | ) | ÷ | ( | x | + | 3 | ) | ) | ( | x | + | 1 | ) | ( | x | + | 2 | ) | + | ( | ( | x | + | 3 | ) | ÷ | ( | x | + | 4 | ) | ) | ( | x | + | 1 | ) | ( | x | + | 2 | ) |
| 4 | x | x | + | 8 | x | + | 4 | x | + | 4 | × | 2 | = | x | x | + | 2 | x | + | x | ( | x | + | 1 | ) | + | 1 | ( | x | + | 1 | ) | + | ( | ( | x | + | 2 | ) | ÷ | ( | x | + | 3 | ) | ) | ( | x | + | 1 | ) | ( | x | + | 2 | ) | + | ( | ( | x | + | 3 | ) | ÷ | ( | x | + | 4 | ) | ) |
| 4 | x | x | + | 8 | x | + | 4 | x | + | 8 | = | x | x | + | 2 | x | + | x | ( | x | + | 1 | ) | + | 1 | ( | x | + | 1 | ) | + | ( | ( | x | + | 2 | ) | ÷ | ( | x | + | 3 | ) | ) | ( | x | + | 1 | ) | ( | x | + | 2 | ) | + | ( | ( | x | + | 3 | ) | ÷ | ( | x | + | 4 | ) | ) |
| 4 | x | x | + | 12 | x | + | 8 | = | x | x | + | 2 | x | + | x | ( | x | + | 1 | ) | + | 1 | ( | x | + | 1 | ) | + | ( | ( | x | + | 2 | ) | ÷ | ( | x | + | 3 | ) | ) | ( | x | + | 1 | ) | ( | x | + | 2 | ) | + | ( | ( | x | + | 3 | ) | ÷ | ( | x | + | 4 | ) | ) |
| 4 | x | x | + | 12 | x | + | 8 | = | x | x | + | 2 | x | + | x | x | + | x | × | 1 | + | 1 | ( | x | + | 1 | ) | + | ( | ( | x | + | 2 | ) | ÷ | ( | x | + | 3 | ) | ) | ( | x | + | 1 | ) |
| 4 | x | x | + | 12 | x | + | 8 | = | x | x | + | 3 | x | + | x | x | + | 1 | ( | x | + | 1 | ) | + | ( | ( | x | + | 2 | ) | ÷ | ( | x | + | 3 | ) | ) | ( | x | + | 1 | ) | ( | x | + | 2 | ) | + | ( | ( | x | + | 3 | ) | ÷ | ( | x | + | 4 | ) | ) |
| 4 | x | x | + | 12 | x | + | 8 | = | x | x | + | 3 | x | + | x | x | + | 1 | x | + | 1 | × | 1 | + | ( | ( | x | + | 2 | ) | ÷ | ( | x | + | 3 | ) | ) | ( | x | + | 1 | ) |
| 4 | x | x | + | 12 | x | + | 8 | = | x | x | + | 3 | x | + | x | x | + | 1 | x | + | 1 | + | ( | ( | x | + | 2 | ) | ÷ | ( | x | + | 3 | ) | ) | ( | x | + | 1 | ) | ( | x | + | 2 | ) |
| 4 | x | x | + | 12 | x | + | 8 | = | x | x | + | 4 | x | + | x | x | + | 1 | + | ( | ( | x | + | 2 | ) | ÷ | ( | x | + | 3 | ) | ) | ( | x | + | 1 | ) | ( | x | + | 2 | ) | + | ( | ( | x | + | 3 | ) | ÷ | ( | x | + | 4 | ) | ) | ( | x | + | 1 | ) |
| 4 | x | x | + | 12 | x | + | 8 | = | x | x | + | 4 | x | + | x | x | + | 1 | + | ( | x | + | 2 | ) | ÷ | ( | x | + | 3 | ) | × | ( | x | + | 1 | ) | ( | x | + | 2 | ) | + | ( | ( | x | + | 3 | ) | ÷ | ( | x | + | 4 | ) | ) |
| Multiply both sides of the equation by: | ( | x | + | 3 | ) |
| 4 | x | x | ( | x | + | 3 | ) | + | 12 | x | ( | x | + | 3 | ) | + | 8 | ( | x | + | 3 | ) | = | x | x | ( | x | + | 3 | ) | + | 4 | x | ( | x | + | 3 | ) | + | x | x | ( | x | + | 3 | ) | + | 1 | ( | x | + | 3 | ) | + | ( | x | + | 2 | ) |
| 4 | x | x | x | + | 4 | x | x | × | 3 | + | 12 | x | ( | x | + | 3 | ) | + | 8 | = | x | x | ( | x | + | 3 | ) | + | 4 | x | ( | x | + | 3 | ) | + | x | x | ( | x | + | 3 | ) | + | 1 | ( | x | + | 3 | ) | + | ( | x | + | 2 | ) |
| 4 | x | x | x | + | 4 | x | x | × | 3 | + | 12 | x | ( | x | + | 3 | ) | + | 8 | = | x | x | x | + | x | x | × | 3 | + | 4 | x | ( | x | + | 3 | ) | + | x | x | ( | x | + | 3 | ) |
After the equation is converted into a general formula, there is a common factor:
( 2x + 5 )
From
2x + 5 = 0
it is concluded that:：
| x1= | - | 5 2 |
Solutions that cannot be obtained by factorization:
x2≈-3.618034 , keep 6 decimal places
x3≈-1.381966 , keep 6 decimal places
There are 3 solution(s).
解程的详细方法请参阅：《方程的解法》