What are the three situations in which a fractional equation has no solution

Fractional equationsThe unsolvable case is:

1. The fractional equation has additional roots.

2. The coefficient of x is not 0. For example, both sides of the equation are multiplied by the simplest common denominator at the same time, and the fractional equation is transformed into an integral equation; If you encounter numbers that are opposite to each other. Don't forget to change the symbols.

After finding the value of the unknown, it is necessary to check the root, because in the process of converting the fractional equation into an integral equation, the range of the value of the unknown is expanded, and the root may be increased.

The ideas for solving fractional equations are summarized as follows:

1) Multiply the simplest common denominator on both sides of the equation at the same time, reduce the denominator, and turn it into an integer equation.

2) Solve this integer equation, which everyone knows.

3) Bring the root of the integer equation into the simplest common denominator to see if the result is zero. If it is zero, it is an additional root of the equation and must be discarded.

4) Write the root of the original equation.

Denominator is zero: When one of the denominators in a fractional equation is zero, the equation has no solution. Because in mathematics, the divisor cannot be zero, a denominator of zero causes the equation to be unsolvable.

For example, for a fractional equation: (x + 3) (x - 2) = 0, when x - 2 = 0, the equation has no solution because the denominator is zero.

The equation is not satisfied:

Some fractional equations have no solution within the range of real numbers, a situation called unsatisfying equations. For example, for a fractional equation: (x 2 - 4) (x - 2) = 0, solving the equation gives x = 2, but when x = 2, the numerator of the equation is 0 and the denominator is not zero, so the equation is not satisfied.

The numerator of the equation is zero, and the denominator is not zero:

Some fractional equations have a numerator of zero, but the denominator is not zero, in which case the equation has a solution. For example, for a fractional equation: (x 2 - 9) (x - 3) = 0, solving the equation gives x = 3, the numerator of the equation is 0, and the denominator is not zero, so the equation has a solution.

Summary: The three situations in which the fractional equation is unsolved are: the denominator is zero, the equation is not satisfied, and the numerator of the equation is zero and the denominator is not zero. When solving fractional equations, you need to be aware of these situations to avoid situations where the divisor is zero or the equation is not satisfied.

A fractional equation is an equation that contains fractions, where at least one unknown number exists, and where the denominator is involved. When solving an equation, there are three cases in which a fractional equation may not be solved:

1.Zero denominator case:

If a denominator in a fractional equation is zero, then the equation will have no solution. Because a denominator of zero causes the fraction in the equation to be undefined or meaningless. Before solving the equation, we need to rule out cases where the denominator is zero.

2.Paradoxical Equation Situation:

If the fractional equation is transformed into a contradictory equation, i.e., the equation becomes an impossible equation, then the equation will have no solution. For example, when the two fractions of a fractional equation are equal, but there is a contradiction between the numerator and the denominator of the fraction on either side of the equation, it is not possible to find a value that satisfies the equation.

3.System Contradiction Situation:

If a fractional equation is part of a multivariate system of equations and is in contradiction with other equations, i.e., two or more equations cannot be satisfied at the same time, then the equation has no solution. When solving a system of equations, we need to check if there are any contradictions in the system of equations.

It is important to note that when solving fractional equations, we usually simplify and deform the equations to make them simpler and easier to solve. In the process of simplification, meaningless solutions or new constraints may be generated, so reasonable checks and verifications are required in the process of solving equations.

In fractional equations, unresolved can occur in the following three situations:

When the denominator is 0: When there is a denominator in the equation and the value of the denominator is 0, the equation has no solution. Because in mathematics, the divisor cannot be 0, a denominator of 0 makes the equation meaningless and therefore unsolved.

Situations where the conversion of an equation to an identity does not hold: Sometimes solving an equation causes some variables to cancel out and end up with an identity. If this identity is not true, i.e., it is wrong, then the equation has no solution.

Contradictory equations: In some cases, equations can lead to contradictory equations, such as 0=1 or 2=3, etc., such equations are impossible to hold, so the equation has no solution.

Example: In the case of an invalid denominator: x + 1 = 5 (x - 2), when x = 2, the denominator is 0, so there is no solution to the equation.

The case where the identity does not hold: 2x + 4 = 2(x + 1) where we get 2x + 2 after 2(x + 1) on the right, and 2x + 4 = 2x + 2 is not true, so the equation has no solution.

In the case of a contradictory equation: 2x + 3 = 2x + 4, where 2x is eliminated from both sides at the same time to get 3 = 4, it is obviously a contradictory equation, so there is no solution to the equation.

In conclusion, when solving fractional equations, you need to pay attention to these situations to avoid getting wrong results or no solution.

regenerate response

1.Denominator is 0: If the denominator of a fractional equation is 0, the equation has no solution. Because in fractions, the divisor cannot be 0.

2.The numerator of a fractional equation is 0 and the denominator is not 0: If the numerator of a fractional equation is 0 and the denominator is not 0, then the equation has no solution. Because 0 divided by any number equals 0, the result of the equation is 0 regardless of the value of the denominator.

3.The numerator and denominator of a fractional equation are both 0: If both the numerator and denominator of a fractional equation are 0, the equation has no solution. Because 0 divided by 0 is undefined and there is no definite result.

The common denominator is 0, which belongs to the category of elementary mathematics.

If there is a problem with the logic of the equation itself, the product of the equation will not interfere with the equation after simplifying or transforming it.

For some undivisible terms, you can appropriately introduce a very small number for division to get an integer.

Hope the above information can help you solve the problem. Please feel free to let me know if you have any other questions.

There are two cases in which a fractional equation is unsolved

One is that after the fractional equation is converted into an integral equation, the integral equation has no solution.

One is that after the fractional equation is converted into an integral equation, the integral equation has a solution, but this solution makes the denominator of the fractional equation 0, which is the root increase.

The generation of root enhancement is caused when the first step of solving the fractional equation, "removing the denominator".

According to the principle of homogeneous solution of equations, both sides of the equation are multiplied (or divided) by the same non-0 number, and the resulting equation is the homogeneous equation of the original equation.

If the number multiplied by both sides of the equation is 0, then the resulting equation is not the same solution as the original equation, and the root obtained is the additional root of the original equation, that is, the original fractional equation has no solution.

Note:

1) Pay attention to the denominator when removing the denominator, do not omit to multiply the integer term.

2) The root is the root of the integral equation formed by removing the denominator from the fractional equation, but it is not the root of the original fractional equation.

3) Root increment so that the simplest common denominator is equal to 0.

4) In a fractional equation, if x is the denominator, then x should not be equal to 0.

Bring x=a into the simplest common denominator, and if x=a makes the simplest common denominator 0, then a is the root of the original equation. If x=a makes the simplest common denominator not zero, then a is the root of the original equation.

If it is less than 0, the equation does not hold, and the common denominator is 0 or 0 when the original equation is brought in

The equation does not hold or the denominator is zero.

Fractional equationsThere is no solution to the situation, whenRootingIt can make the simplest publicDenominatorWhen it is equal to 0, there is no solution to the equation, and when the root is the root of the whole equation obtained after removing the denominator, the equation has no solution. The basic idea of solving fractional equations is to convert fractional equations into integral equations and then solve them, which reflects the idea of transformation.

The meaning of fractional equationsFractional equation is a kind of equation, which refers to a rational equation with an unknown number or an unknown integer in the denominator, and this part of the knowledge belongs to elementary mathematical knowledge. There is at least one rational equation with an unknown fraction on either side of the equal sign. Multiplying the lowest common denominator of each fraction in the equation by both sides of the equation can convert the fractional equation into an integral equation to solve, but it may produce root addition, so the root must be checked.

The unsolved fractional equation means that no matter what value is taken, the equal sign of the fractional equation cannot be satisfied, and the two sides are equal. The equation has no solution when the root is added so that the simplest common denominator is equal to 0, and when the root is the root of the integer equation obtained by removing the denominator, the equation has no solution.

The unsolved fractional equation means that no matter what value is taken, the equality of both sides of the fractional equation cannot be satisfied, and there are two main situations in which the fractional equation is unsolved

1. After the original fractional equation is multiplied by the simplest common denominator on both sides of the equal sign at the same time to reduce it to an equation equation, the equation has no solution;

2. After the fractional equation is converted into an equation equation, the integral equation has a solution, but this solution makes the denominator of the original fractional equation 0, and this solution is called the root addition of the fractional equation.

If the property of non-solution of fractional equations can be correctly applied in the actual problem solving, it will help to effectively improve the efficiency of problem solving, understand the problem more clearly, and solve other problems.

When solving fractional equations:

Removing the denominator so that the solution of the integer equation obtained after the loss is known may make the denominator in the original equation zero, so the solution of the integer equation should be substituted into the simplest common denominator, and if the value of the simplest common denominator is not zero, it is the solution of the equation.

If the simplest common denominator is equal to 0, the root is an incremental root. Otherwise, this root is the root of the hollow branch primitive equation. If the solved roots are all incremental roots, then there is no solution to the original equation.

Fractional equation has no solution:

1. The fractional equation has additional roots.

2. The coefficient of x is not 0.

Such as: <>

Both sides of the equation are multiplied by the simplest common denominator at the same time, and the fractional equation is reduced to an integer.

Equation; If you encounter numbers that are opposite to each other. Don't forget to change the symbols.

Lowest common denominator: The coefficient is the lowest common multiple.

The unknown number is taken to the highest power; Factors that appear.

Take the highest power. ）