Solution idea

1, simplify first, and multiply both sides of the equation by the common factor (1+x) 5.

2. Set y= 1+x to simplify the sorting equation and extract the common factor (y? +y+ 1) and get (y? +y+ 1)( 10y？ - 1 1)=0

3. Solve the equation, get the y value, and then get the x value.

Solution process

Topic knowledge point

1, the steps to solve the fractional equation.

1) denominator

Multiply both sides of the equation by the simplest common denominator at the same time, and turn the fractional equation into an integral equation; If you encounter the opposite number. Don't forget to change the symbol.

(The simplest common denominator: ① coefficient takes the least common multiple ② unknown takes the highest power ③ the factor that appears takes the highest power)

2) Move the project

Move the term, if there are brackets, first remove the brackets, pay attention to the sign change, merge similar terms, change the coefficient into 1, and find the unknown value;

3) Root inspection

After finding the value of the unknown quantity, it is necessary to check the root, because in the process of transforming the fractional equation into the whole equation, the range of the unknown quantity is expanded, which may increase the root.

When finding the root, substitute the root of the whole equation into the simplest common denominator. If the simplest common denominator is equal to 0, this root is an added root. Otherwise, this root is the root of the original fractional equation. If all the roots are augmented, the original equation has no solution.

If the score itself is divided, it should also be substituted into the test.

When solving an application problem with a column fraction equation, it is necessary to check whether the obtained solution meets the equation and the meaning of the problem.

Generally speaking, when solving fractional equations, the solution of the whole equation after removing the denominator may make the denominator in the original equation zero, so we should substitute the solution of the whole equation into the simplest common denominator. If the value of the simplest common denominator is not zero, it is the solution of the equation.

2. Real number roots. Real root means that the solution of the equation is a real number solution. Real numbers include positive numbers, negative numbers and 0. Some equations have added roots and need to be tested before they can be discarded.

The real root means that the solution of the equation is a real number, and the real root is often called the real root.

1. Real numbers include positive numbers, negative numbers and 0.

1) Positive numbers include: positive integer and positive fraction.

2) Negative numbers include negative integers and negative fractions.

2. Real numbers include rational numbers and irrational numbers.

3. Plural roots. Complex root means that the solution of the equation is a complex solution. Numbers in the form of a+bi (both a and b are real numbers) are complex numbers, where a is called the real part, b is called the imaginary part, and I is the imaginary part. Complex numbers are usually represented by z, that is, z=a+bi. When the imaginary part b of z = 0, z is a real number. When the imaginary part b≠0 and the real part a = 0 of z, z is often called pure imaginary number.

Complex number field is an algebraic closure of real number field, that is, any polynomial with complex coefficients always has roots in complex number field.