Hello, why does it come about?

For fractional equations, when the value of the denominator is zero in the fractional equation, it is meaningless, so the fractional equation does not allow the unknowns to take those values that make the value of the denominator zero, that is, the fractional equation itself implies the condition that the denominator is not zero. When a fractional equation is converted into an integral equation, this restriction is removed, in other words, the range of unknowns in the equation is expanded, and if the root of the transformed integral equation happens to be a value other than the allowable value of the unknowns of the original equation, then the root addition will occur.

Because in the process of removing the denominator, we have multiplied the simplest common denominator for each item [that's what it is called], but we omitted the case where the simplest common denominator is 0, and once it is 0, the root increase is generated.

When solving fractional equations, Sone is created because removing the denominator requires multiplying both sides of the equation by an algebraic equation that may be 0.

Look at the conditions set by the question, or the limitations of the formula itself. For example, when you want to solve a fractional equation, you usually have to multiply polynomials (there are no restrictions at this time, so here's the problem), and use the root penetration method to solve it, in this case, it is important to note that the original denominator is not zero. Then this solution is to increase the roots.

Occurs when you transfer a similar item or when you multiply and divide by one.

Because the denominator of a fraction has a finite range and cannot be equal to 0, while the usual integer can be equal to 0 without a finite

Root addition refers to the root that does not meet the problem conditions after the equation is solved. Quadratic equations, fractional equations, and other equations that produce multiple solutions may have additional roots under certain conditions.

In the process of converting the fractional equation into an integral equation, the condition for solving the fractional equation is that the denominator of the original equation is not zero. If the root of an integral equation is such that the simplest common denominator is 0, (the root makes the integral equation true, and in the fractional equation the denominator is 0), then this root is called the incremental root of the original fractional equation. <>

Root test of the equation

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 number is expanded, and the root addition may be generated. If the simplest common denominator is equal to 0, the root is an incremental root. Otherwise this root is the root of the original fractional equation.

If the solved roots are all incremental roots, then there is no solution to the original equation.

If the fraction itself is divided, it should also be substituted for inspection. When solving a problem with a column fractional equation, it is necessary to check not only whether the solution satisfies the equation, but also whether it conforms to the meaning of the problem. In general, when solving a fractional equation, the solution of the integral equation obtained after removing the denominator may make the denominator in the original equation zero, so the solution of the integral 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.

1. When the equation is deformed, sometimes a root that is not suitable for the original equation may be generated, which is called the additional root of the original equation.

2. Reasons for the occurrence of root addition:

1) For the fractional equation, it is meaningless when the value of the denominator is zero in the fractional equation, so the fractional equation does not allow the unknown to take those values that make the value of the denominator zero, that is, the fractional equation itself implies the condition that the denominator is not zero.

2) When the fractional equation is converted into an integral equation, this restriction removes the masking, in other words, the range of values of the unknowns in the equation is expanded, and if the root of the transformed integral equation happens to be a value other than the allowable value of the unknowns of the original equation, then the root increment will occur.

3) Both sides of the fractional equation are multiplied by the simplest common denominator fractional equation as an integral equation, and the allowable value of the unknown is expanded, so the solution of the fractional equation is prone to coarse and eliminate the roots.

When solving a fractional equation, when root enhancement occurs.

When students solve an equation, if there is an increase in roots, it is often caused by violating the principle of the same solution of the equation or being careless when deforming the equation.

1.For example, if you multiply both sides of the equation x 2=0 to form x(x 2)=0, the new equation will have one more root x=0 than the original equation This is because multiplying x on both sides of the equation is equivalent to multiplying 0 by both sides of the original equation (0 is suitable for the new equation), which is contrary to the principle of homogeneous solution.

2.When solving a fractional equation, removing the denominator does not necessarily result in root addition. When deforming a fractional equation, it is often first transformed into an integral equation, so that both sides of the fractional equation are multiplied by the lowest common multiple of each denominator, which may not violate the principle of the same solution, or it may violate the principle of the same solution, such as multiplying both sides of the equation by x, deforming x 2 = 1, the new equation has a root x = 3, which is also the root of the original equation.

x=3 is not an additional root of the original equation, because x multiplied on both sides of the equation is a non-zero number equivalent to 3, which does not violate the principle of homogeneity.

To determine the root increase, just substitute the root of the new equation into the simplest common denominator multiplied on both sides of the original equation when removing the denominator, and see if it is 0, which is the root increase.