Strategies and methods of simplifying the calculation of quadratic root quadratic root is a difficult content in junior high school mathematics teaching. After mastering the concepts and properties related to quadratic roots, readers generally follow the following practices when simplifying and calculating quadratic roots: ① First, simplify the quadratic roots in the formula appropriately; ② The multiplication of quadratic roots can refer to polynomial multiplication, and the division of quadratic roots with formula (,) ③ is usually written as a fraction first. And then operate through the denominator. ④ The addition and subtraction of quadratic roots is similar to the addition and subtraction of polynomials, that is, on the basis of simplification, brackets are removed and similar terms are merged. ⑤ The operation result is generalized to the simplest quadratic root. The common skills and methods of simplifying secondary roots are important contents of secondary roots teaching. For the simplification of quadratic roots, besides the basic concepts and operation rules, we must also master some special methods and skills. You will get twice the result with half the effort. The following examples are classified and analyzed. 1. Formula example 1 calculation ①; ② Solution ① Original Formula ② After annotating the solution of the original formula, the "complete square formula" and "square difference formula" are used in the above solution, which makes the calculation simpler. 2. Observation of the calculation of the characteristic method example 2: Method guidance If the calculation is directly based on the nature of the root formula, the denominator needs to be rationalized twice, which is quite troublesome to calculate. Observing the numerator and denominator in the original formula, we can find that the numerator is obtained by multiplying the items in the denominator, so the simple solution can be as follows: (2) (1) Method leads to two factors in the denominator. Multiplying by two physical and chemical factors will make the three factors of the molecule equal, and the calculation will be very complicated. If the two factors in the denominator are added together, the following solutions can be found: solving ① the original method leads ② the denominator can be directly materialized, and then simplified. But it is not difficult to find that if the coefficient in ② is "65438", ② can be answered as follows: Solve ② the original formula 3. After simplifying the solution of the original formula by using example 4 of collocation method, it is noted in the notes that this is an arithmetic root, and it must be non-negative after drawing the root, which obviously cannot be equal to ""4. Example 5 of the plane method simplifies the solution \u ∴. The comments after the solution are all related to this * * * yoke root formula. Generally, it can be simplified by flattening. 5. Constant deformation formula Example 6. Simplify method guidance. If it is directly expanded, the calculation is complicated. If a formula is used, the operation will be simplified. Solve the original equation 6. Constant substitution law 7. Simplify the original formula. Example 8. Simplify the denominator to get the original formula. Example 9. Simplify method guidance. If the denominator is directly materialized, it will be very complicated. But it is not difficult to find that the numerator of each fraction is equal to the sum of the two factors of the denominator, so we have the following simple solution: solve the original formula 8. Construct dual formula. Example 10 simplifies the dual formula, so there is no, then, the original formula 9. Simplify the solution step by step ∵ and ∴ After solving the original formula, the commentary should adopt the problem of simplifying multiple roots. Simplify by layer. 10. From right to left, simplify the example item by item. 1 1 Simplifies the original formula from right to left, so it is simplified from right to left. After solving the original formula, it is the key to solve this problem to comment on the square difference formula and the overall idea. The square difference formula is used to remove multiple radicals layer by layer and simplify them item by item. Without skilled skills, it is difficult to achieve the goal of simplifying the complex. The common method of comparing the size of quadratic root is very skillful, and it is also very skillful to compare the size of two irrational numbers (that is, quadratic root) without seeking approximate value, which is the difficulty of junior middle school students. However, mastering some common methods is of great help and promotion to their learning. 1. Example of radical deformation method 1 Compare the size solutions of two quadratic roots and convert them into:, ∴ that is, the notes after the solution. The basis of this solution is: when, when, ①, then; 2 if, then 2. Example 2: Comparison and evaluation of large and small solutions The basis of this method is: when, when, if, then, if. 3. Denominator rationalization method uses denominator rationalization and uses the size of the molecule to judge the size of its reciprocal. Example 3: Compare the size solutions of and. We usually want to make it molecular, and judge its reciprocal by the size of denominator. Comparative slurry solutions of Example 4 are ∵ and ∴. and 5. Basic properties of the equation. The dimensional solution of Comparative Example 5 is1:1,and ∴ is the note after the solution. This solution makes use of the following two properties: ① After the same numbers are added, the size relationship between the two numbers remains unchanged. ② The relationship between nonnegative basis and its quadratic power is consistent. Solution 2 Multiply them by the physical and chemical factors of these two numbers respectively, and the solution is ∵∴∴∴∴. The basis of this solution is that the size relationship remains unchanged after both numbers are multiplied by the same positive number. 6. The slurry solution of Example 6 was compared by the medium value transfer method. (2) Comparison between Example 7 and Slurry Solution ∵∴ 8. There is also a comparison method corresponding to the difference comparison method, that is, the quotient comparison method, which uses the following properties: when, when, then: ①; (2) Comparison of the sum size of Example 8. It is mentioned in the comments after the solution that irrational numbers can often be solved by various methods compared with the size of roots. Sometimes a variety of methods are needed, among which root deformation method and flat method are the most basic. Only by analyzing specific problems can we solve the correct results in the best way.