Solution: Example: (1) x/(x+1) = 2x/(3x+3)+1.
Multiply both sides by 3(x+ 1)
3x=2x+(3x+3)
3x=5x+3
2x=-3
x=-3/2
The fractional equation should be tested.
X=-3/2 is the solution of the equation.
(2)2/x- 1=4/x^2- 1
Multiply both sides by (x+ 1)(x- 1)
2(x+ 1)=4
2x+2=4
2x=2
x= 1
The fractional equation should be tested.
After testing, x= 1 makes the denominator 0, which is an added root.
So the original equation 2/x-1= 4/x 2-1.
No solution. Remember to check whether it is a zenggen when solving fractional equations.
What is the significance of adding roots to fractional equations?
When a fractional equation is transformed into an integral equation, do you "multiply both sides by xxxx at the same time"
This change is the premise of the same solution change, but your xxxx is not equal to 0.
But sometimes, that xxxx is equal to 0, which can just satisfy the whole equation, but it should not be the solution of the fractional equation. This is rooting.
How to find the value of letters in fractional equation by increasing roots
1, divided by the denominator to get the whole equation,
2. Substitute the x value with denominator of 0 into the whole equation.
3. Get the equation of letters and solve it to get the value of letters.
How to find the fractional equation with increasing roots
I hope I can help you!
Rooting: when naming a fractional equation, sometimes roots that are not suitable for the original equation may be generated, which is called rooting of the original equation.
Analysis: Because increasing roots may occur when solving fractional equations, it is necessary to solve fractional equations.
Inspection method:
(1) Method for checking whether the root is increased:
Usually, the root obtained is substituted into the simplest common denominator after denominator removal to see whether its value is 0, so that the root with the simplest common denominator of 0 is the additional root of the original equation and must be discarded. The simplest root whose common denominator is not zero is the root of the original equation. (This test must be written into the step of solving equations, a necessary step)
(2) Check whether the equation you solved is correct, and substitute the unknown values into the left and right sides of the equation to see if the left and right sides are equal.
What is rooting? Why does the solution of fractional equation increase roots?
(1) Root-increasing: a mathematical term, which means that in the process of transforming a fractional equation into an integral equation, if the root of the integral equation makes the simplest common denominator 0, then this root is called root-increasing of the original fractional equation.
For example:
x/(x-2)-2/(x-2)=0
Solution: denominator, x-2=0.
x=2
But X=2 makes the denominator equal to 0 (meaningless), so X=2 is an incremental root.
(2) Because the range of independent variables is enlarged after the denominator is removed, that is to say, the numbers that are not in the range may also be the solution of the whole equation after the denominator is removed, so there may be root increase in the process of solving the fractional equation after the denominator is removed.