I. Numbers and algebra

I. Numbers and formulas:

1, rational number

Rational Numbers: ① Integer → Positive Integer /0/ Negative Integer

② Score → Positive/Negative Score

Number axis: ① Draw a horizontal straight line, take a point on the straight line to represent 0 (origin), select a certain length as the unit length, and specify the right direction on the straight line as the positive direction to get the number axis. ② Any rational number can be represented by a point on the number axis. (3) If two numbers differ only in sign, then we call one of them the inverse of the other number, and we also call these two numbers the inverse of each other. On the number axis, two points representing the opposite number are located on both sides of the origin, and the distance from the origin is equal. The number represented by two points on the number axis is always larger on the right than on the left. Positive numbers are greater than 0, negative numbers are less than 0, and positive numbers are greater than negative numbers.

Absolute value: ① On the number axis, the distance between the point corresponding to a number and the origin is called the absolute value of the number. (2) The absolute value of a positive number is itself, the absolute value of a negative number is its reciprocal, and the absolute value of 0 is 0. Comparing the sizes of two negative numbers, the absolute value is larger but smaller.

Operation of rational number:

Addition: ① Add the same sign, take the same sign, and add the absolute values. ② When the absolute values are equal, the sum of different symbols is 0; When the absolute values are not equal, take the sign of the number with the larger absolute value and subtract the smaller absolute value from the larger absolute value. (3) A number and 0 add up unchanged.

Subtraction: Subtracting a number equals adding the reciprocal of this number.

Multiplication: ① Multiplication of two numbers, positive sign of the same sign, negative sign of different sign, absolute value. ② Multiply any number by 0 to get 0. ③ Two rational numbers whose product is 1 are reciprocal.

Division: ① Dividing by a number equals multiplying the reciprocal of a number. ②0 is not divisible.

Power: the operation of finding the product of n identical factors A is called power, the result of power is called power, A is called base, and N is called degree.

Mixing order: multiply first, then multiply and divide, and finally add and subtract. If there are brackets, calculate first.

2. Real numbers

Irrational number: Infinitely circulating decimals are called irrational numbers.

Square root: ① If the square of a positive number X is equal to A, then this positive number X is called the arithmetic square root of A. If the square of a number X is equal to A, then this number X is called the square root of A. (3) A positive number has two square roots /0 square root is 0/ negative number without square root. (4) Find the square root of a number, which is called the square root, where a is called the square root.

Cubic root: ① If the cube of a number X is equal to A, then this number X is called the cube root of A. ② The cube root of a positive number is positive, the cube root of 0 is 0, and the cube root of a negative number is negative. The operation of finding the cube root of a number is called square root, where a is called square root.

Real numbers: ① Real numbers are divided into rational numbers and irrational numbers. ② In the real number range, the meanings of reciprocal, reciprocal and absolute value are exactly the same as those of reciprocal, reciprocal and absolute value in the rational number range. ③ Every real number can be represented by a point on the number axis.

3. Algebraic expressions

Algebraic expression: A single number or letter is also an algebraic expression.

Merge similar items: ① Items with the same letters and the same letter index are called similar items. (2) Merging similar items into one item is called merging similar items. (3) When merging similar items, we add up the coefficients of similar items, and the indexes of letters and letters remain unchanged.

4. Algebraic expressions and fractions.

Algebraic expression: ① The algebraic expression of the product of numbers and letters is called monomial, the sum of several monomials is called polynomial, and monomials and polynomials are collectively called algebraic expressions. ② In a single item, the index sum of all letters is called the number of times of the item. ③ In a polynomial, the degree of the term with the highest degree is called the degree of this polynomial.

Algebraic expression operation: when adding and subtracting, if you encounter brackets, remove them first, and then merge similar items.

Power operation: AM+AN=A(M+N)

(AM)N=AMN

(A/B)N=AN/BN division.

Multiplication of algebraic expressions: ① Multiply the monomial with the monomial, respectively multiply their coefficients and the power of the same letter, and the remaining letters, together with their exponents, remain unchanged as the factors of the product. (2) Multiplying polynomial by monomial means multiplying each term of polynomial by monomial according to the distribution law, and then adding the products. (3) Polynomial multiplied by polynomial. Multiply each term of one polynomial by each term of another polynomial, and then add the products.

There are two formulas: square difference formula/complete square formula.

Algebraic division: ① monomial division, which divides the coefficient and the power of the same base as the factor of quotient respectively; For the letter only contained in the division formula, it is used as the factor of quotient together with its index. (2) Polynomial divided by single item, first divide each item of this polynomial by single item, and then add the obtained quotients.

Factorization: transforming a polynomial into the product of several algebraic expressions. This change is called factorization of this polynomial.

Methods: Common factor method, formula method, grouping decomposition method and cross multiplication were used.

Fraction: ① Algebraic expression A is divided by algebraic expression B. If the divisor B contains a denominator, then this is a fraction. For any fraction, the denominator is not 0. ② The numerator and denominator of the fraction are multiplied or divided by the same algebraic expression that is not equal to 0, and the value of the fraction remains unchanged.

Operation of fraction:

Multiplication: take the product of molecular multiplication as the numerator of the product, and the product of denominator multiplication as the denominator of the product.

Division: dividing by a fraction is equal to multiplying the reciprocal of this fraction.

Addition and subtraction: ① Add and subtract fractions with the same denominator, and add and subtract molecules with the same denominator. ② Fractions with different denominators shall be divided into fractions with the same denominator first, and then added and subtracted.

Fractional equation: ① The equation with unknown number in denominator is called fractional equation. ② The solution whose denominator is 0 is called the root increase of the original equation.

Equations and inequalities

1, equation and equation

Unary linear equation: ① In an equation, there is only one unknown, and the exponent of the unknown is 1. Such an equation is called a one-dimensional linear equation. ② Adding or subtracting or multiplying or dividing (non-0) an algebraic expression on both sides of the equation at the same time, the result is still an equation.

Steps to solve a linear equation with one variable: remove the denominator, shift the term, merge the similar terms, and change the unknown coefficient into 1.

Binary linear equation: An equation that contains two unknowns and whose terms are 1 is called binary linear equation.

Binary linear equations: The equations composed of two binary linear equations are called binary linear equations.

A set of unknown values suitable for binary linear equation is called the solution of this binary linear equation.

The common * * * solution of each equation in a binary linear system of equations is called the solution of this binary linear system of equations.

Methods of solving binary linear equations: substitution elimination method/addition and subtraction elimination method.

Quadratic equation of one variable: an equation with only one unknown and the highest coefficient of the unknown term is 2.

1) The relation of quadratic function of quadratic equation in one variable.

Everyone has studied quadratic function (parabola) and has a deep understanding of it, such as solution and representation in images. In fact, the quadratic equation of one variable can also be expressed by quadratic function. In fact, the quadratic equation of one variable is also a special case of quadratic function, that is, when y is 0, it constitutes the quadratic equation of one variable. Then, if expressed in a plane rectangular coordinate system, the quadratic equation of one variable is the intersection of the X axis in the image and the quadratic function. Which is the solution of the equation.

2) Solution of quadratic equation in one variable.

As we all know, a quadratic function has a vertex (-b/2a, 4ac-b2/4a), which is very important. Remember, as mentioned above, the quadratic equation with one variable is also a part of the quadratic function, so he also has his own solution, and he can find all the solutions of the quadratic equation with one variable.

(1) matching method

Using this formula, the equation is transformed into a complete square formula and solved by direct Kaiping method.

(2) Factor decomposition method

Select the common factor, apply the formula, and cross multiply. The same is true for solving quadratic equations with one variable. Using this, the equation can be solved by several products.

(3) Formula method

This method can also be used as a general method to solve quadratic equations with one variable. The roots of the equation are x 1 = {-b+√ [B2-4ac]}/2a, and x2 = {-b-√ [B2-4ac]}/2a.

3) the step of solving a quadratic equation with one variable:

(1) Matching method steps:

First, the constant term is moved to the right of the equation, then the coefficient of the quadratic term is changed to 1, and the square of half the coefficient of 1 is added, and finally the complete square formula is obtained.

(2) The steps of factorization:

Turn the right side of the equation into 0, and then see if you can extract the common factor, formula (here refers to the formula in factorization) or cross multiplication, and if you can, turn it into the form of product.

(3) Formula method

Simply substitute the coefficient of quadratic equation into a variable, where the coefficient of quadratic term is a, the coefficient of linear term is b, and the coefficient of constant term is c.

4) Vieta theorem

Understand with Vieta theorem, Vieta theorem in a quadratic equation, the sum of two roots =-b/a, the product of two roots = c/a.

It can also be expressed as x 1+x2 =-b/a, and x1x2 = c/a. By using Vieta's theorem, we can find out the coefficients in a quadratic equation with one variable, which is very common in the topic.

5) The case of the root of a linear equation with one variable

Using the discriminant of roots to understand, the discriminant of roots can be written as "Delta" and read as "Tune ta", and Delta = B2-4ac can be divided into three situations:

I am △ > 0, a quadratic equation with one variable has two unequal real roots;

II When △=0, the quadratic equation of one variable has two identical real roots;

Three dang △

2. Inequality and unequal groups

Inequality: ① When the symbol > = 0 is used, it passes through quadrant124; When k > 0 and b < 0, pass through quadrant134; When k > 0 and b > 0, pass through quadrant 123. ④ When k > 0, y value increases with the increase of x value, and when x < 0, y value decreases with the increase of x value.

I hope it helps you.