Note: (1) If this condition does not hold, it is not a quadratic radical;

(2) is an important non-negative number, that is; 0.

2. Important formula: (1), (2)

3. Arithmetic square root of product:

The arithmetic square root of the product is equal to the product of the arithmetic square root of each factor in the product;

4. The multiplication rule of quadratic root:

5. Square root comparison method:

(1) Compare sizes with approximate values;

(2) Move the coefficient of the quadratic root into the quadratic root number, and then compare the sizes;

(3) Square separately and then compare the sizes.

6. The arithmetic square root of quotient:

The arithmetic square root of quotient is equal to the arithmetic square root of divisor divided by the arithmetic square root of divisor.

7. The division rule of quadratic root:

( 1); (2);

(3) The method to make the denominator rational is to multiply the numerator and denominator of the fraction by the rational factor of the denominator respectively, so that the denominator becomes an algebraic expression.

8. The simplest quadratic root:

(1) A quadratic root that satisfies the following two conditions is called the simplest quadratic root. ① The factor of the root sign is an integer and the factor is an algebraic expression. ② The radical sign does not contain all the factors that can be turned on.

(2) In the simplest quadratic root, the root number cannot contain decimals and fractions, and the number of letter factors is less than 2, excluding the denominator;

(3) When simplifying the quadratic square root, it is often necessary to decompose the square root into factors or factorizations first;

(4) The final result of quadratic formula calculation must be simplified to the simplest quadratic formula.

10. Homogeneous secondary roots: after several secondary roots are transformed into the simplest secondary roots, if the number of roots is the same, these secondary roots are called homogeneous secondary roots.

12. Mixed operation of quadratic roots:

The' mixed operation' of (1) quadratic roots includes six algebraic operations: addition, subtraction, multiplication, division, multiplication and root. All the formulas and algorithms in the range of rational numbers that I have learned before are applicable to the mixed operation of quadratic roots.

(2) In general, the quadratic roots must be simplified properly before operation, for example, they can only be merged after being converted into similar quadratic roots; Sometimes it is easier to convert the division operation into the denominator in a rational or reduced way. Use multiplication formulas, etc.

Chapter 22 One-variable quadratic equation

1. The general form of the unary quadratic equation: 0, ax2+bx+c=0 is called the general form of the unary quadratic equation. When studying the related problems of quadratic equation with one variable, most of the exercises must be converted into general forms before A and B can be determined, where A, B and C may be concrete numbers, algebraic expressions of undetermined letters or concrete formulas.

2. Solution of quadratic equation with one variable: The four solutions of quadratic equation with one variable require flexible application, among which the direct leveling method is simple, but its application scope is small; Although the formula method has a wide range of applications, it is complicated in calculation and prone to calculation errors. Factorization is the first choice for its wide application and simple calculation. Matching method is less used.

3. Discriminant formula of the root of a quadratic equation with one variable: when ax2+bx+c=00), =b2-4ac is called discriminant of the root of a quadratic equation with one variable. Please note the following equivalent propositions:

0= there are two unequal real roots; =0= There are two equal real roots; 0= No real root;

4. The average growth rate problem-one of the application problems (let the growth rate be x):

(1) The first year is A, the second year is a( 1+x), and the third year is a( 1+x)2.

(2) We often use the following equation relationship to form an equation: the third year = the third year or the first year+the second year+the third year = and.