Addition operation of rational numbers

Two numbers with the same sign are added, and the absolute value is added with the same sign.

Different symbols increase or decrease, and large numbers determine symbols.

Sum the opposite numbers, and the result is zero. Be sure to remember it.

Note that "big" minus "small" refers to the absolute value.

Subtraction operation of rational number

Minus positive equals to add negative, and reducing burden equals to add positive.

Symbolic law of rational number multiplication operation

The sign of the same sign is negative and the product of a term is zero.

Combine similar terms

When it comes to merging similar projects, don't forget the rules.

Only the algebraic sum of the coefficients is found, and the letter index remains unchanged.

Rules for deleting and adding brackets

The key to deleting brackets or adding brackets depends on the connection number.

There is a plus sign in front of the expansion symbol, and the bracket-invariant symbol is deleted.

There is a minus sign in front of the parentheses, and it will change when you add it.

solve an equation

Known and unknown causes are separated, and this separation should be done by moving.

Shift addition, subtraction, addition, multiplication, division and multiplication.

formula for the difference of square

The sum of two numbers multiplied by the difference of two numbers is equal to the square of the difference of two numbers.

Product and difference are two nouns, but complete square is not.

Perfect square trinomial

The sum or difference of the squares of two numbers is expanded into three terms.

The first square and the last square, the first and the last two in the middle.

Add and add the square of the sum, subtract first and then add the square of the difference.

Perfect square trinomial

The first square is the last square, and the second square is the first and last one in the center.

The squares of sum are added and then added, and the squares of difference are subtracted and then added.

Solve a linear equation with one variable

Remove the denominator first and then the brackets, and remember the sign of the shifted item.

The coefficient "1" is not enough for the merger of similar items.

To obtain the unknown quantity, the value must be checked and replaced.

Solve a linear equation with one variable

Remove the denominator first and then the brackets, and move items to merge similar items.

The coefficient of 1 is not ready yet, and the calculation is not in vain.

Factorization and multiplication

The product of sum and difference is multiplication, and multiplication itself is operation.

Product sum and difference are decomposition, and factorization is not operation.

factoring

Don't be afraid of factorization just because the square signs of two formulas are different.

Multiply the sum of two cardinality by the difference of two cardinality, and the decomposition result is it.

The square sign of the two formulas is the same, and the bottom product is twice the center.

Can factorization? There is an article on the symbol.

The same difference is squared first and signed.

The positive and negative sign of the same law is negative, and the difference is plus the power sign.

factoring

One mention, two sets, three sets, cross multiplication is also counted.

None of the four methods works, so we have to split the items and add items to reorganize.

It is hopeless to try to find the root, exchange elements or calculate the remainder in recombination.

A variety of methods can be flexibly selected, and the result of continuous multiplication is the basis.

If the same type of multiplication occurs, this ability means remembering.

be filled/suffused/brimming with

One mention (common factor) and two sets (formula sets)

factoring

One mention, two episodes, three groups, and the cross product of the root is also counted.

None of the five methods works, so we have to split the items and add items for reorganization.

Suit the remedy to the case, seek accuracy in stability, and the result of serial multiplication is the foundation.

Factorization of quadratic trinomial

Think completely flat first, then cross.

Neither method works. Try root decomposition.

Ratio and proportion

Division of two numbers is also called ratio, and equality of two numbers is also called ratio.

The outer product equals the inner product, and the equal product can be divided into eight proportions.

When the internal and external items are exchanged separately, it is called comparison.

The simultaneous exchange of internal and external terms is called inverse ratio.

The ratio before and after the term is constant, which is called the combined ratio.

The difference between the preceding item and the latter item is the ratio.

The sum of two items is not as good as two items, and the proportion is equal.

The sum of the preceding paragraph is equal to the sum of the following items, and the proportion remains unchanged.

Solution ratio

Outer product and inner product, list equations and solve them.

Find the ratio

There are many ways to find the ratio from known values.

Flexible use of the property of ratio seven, variable substitution is also very popular.

It's a good idea to destroy Yolanda, and all roads lead to the same goal.

Positive proportion and inverse proportion

The agreed variables are directly proportional and the product variables are inversely proportional.

Positive proportion and inverse proportion

The change process quotient is certain, and the two variables are proportional.

The product of the changing process is constant, and the two variables are inversely proportional.

Judge that four numbers are proportional.

Whether the four numbers are proportional or not is sorted in ascending and descending order first.

The product of two ends is equal to the intermediate product, and four numbers must be proportional.

Judge that the four formulas are proportional.

Whether the four formulas are proportional or not, the order is "Mr" and "Mr".

The two-terminal product is equal to the intermediate product, and the four formulas can be proportional.

mean proportional

Among the four proportional terms, the external term is the same.

Sometimes the internal items will be the same, and the intermediate items in the proportion are essential.

The middle term in the proportion is very important and will be encountered in many occasions.

Among the four proportional terms, many external terms are the same.

Sometimes the internal items will be the same, and the items in the proportion will appear.

The same number, square, different products, there is nowhere to escape in proportion.

Radical and irrational

An algebraic expression representing a square root can be called a radical.

The radical form is different from the irrational form, and its opening mode is not limited.

Only when there are letters in the opened way can it be called unreasonable.

Unreasonable forms are radical forms, which are distinguished by signs.

There are letters on the way to be opened, which can also be called unreasonable.

Find domain name

Four principles should be paid attention to when seeking the domain.

Negative numbers cannot be squared, and zero denominator is meaningless.

Refers to the positive number at the bottom of the fraction, and zero has no power of zero.

Constraints are not unique and satisfy multiple inequalities.

If you want to pass the customs, you should pay attention to four principles.

Negative numbers cannot be squared, and zero denominator is meaningless.

Fractional index has a positive base, and zero has no zero power.

Constraints are not unique, solve the inequality group.

Solving one-dimensional linear inequality

Remove the denominator first and then the brackets, and move items to merge similar items.

The coefficient of "1" is particular, and the same multiplication and division method needs to change direction.

Remove the denominator first, then the brackets, and don't forget to change the symbol when moving the item.

When similar items are merged, the coefficient is "1".

There is no obstacle to the same multiplication and division, and the same multiplication and division also changes sign.

Solving a system of linear inequalities with one variable

Greater than the head and less than the tail, find the middle of different sizes.

There is no solution to the size, and all four situations are coming.

Take two sides in the same direction and take the middle in the opposite direction.

There is no element in the middle, no solution.

Kindergarten children are responsible, (just like the younger ones)

Nursing homes are proud of being old.

There are no old people or young people in the barracks.

All schemes, large and small, are empty. (There are no small and large wow)

Solve a quadratic inequality in one variable

First, it is transformed into a general formula, and the second station of the constructor.

If the discriminant value is not negative, the horizontal axis of the curve has an intersection point.

A is opening it. If it is greater than zero, take both sides.

If the algebraic expression is less than zero, the intersection of solution sets.

If the equation has no real root, the big zero solution in the mouth is all.

If it is less than zero, there is no solution, and the opening is just the opposite.

Factorization with square difference formula

There is a way to decompose two square terms with different symbols.

Multiply the sum of two cardinality by the difference of two cardinality, and the decomposition result is it.

Factorization of complete square formula

Two square terms are at both ends, and the bottom product is twice that of the middle.

The sum of the same base and the square, the sum of the reciprocal of all negative numbers and the square.

Divided into two base difference squares, the square product should be negative.

Both sides are negative, the middle is positive, and the square of cardinal difference is inverse.

One side after another, the bottom product is twice that of the middle.

Three plus two bases sum square, total subtraction and square reciprocal.

Divided into two squares with base difference, with positive product and negative at both ends.

If both sides are negative, the middle is positive, and the square of the base difference is opposite.

Solving quadratic equation with one variable by formula method

Solve the equation with a formula, and first convert it into a general formula.

Then follow the adjustment coefficient to make it the simplest ratio.

Determine the parameter abc and calculate the discriminant of the equation.

If there are any real roots, the ratio of discriminant value to zero will be known.

There is a real root setting formula, but there is no real root to talk about.

Solving quadratic equation with one variable by conventional collocation method

First separate the left and right, and then binarize "1".

A series is folded in half and then squared, and the two sides add up.

Divide left and right, and solve the problem directly.

This solution is called a formula, so you should practice more when solving the equation.

Indirect collocation method for solving quadratic equation with one variable

The known unknowns are separated first, and then factorized.

The adjustment coefficient is reciprocal, and the sum-difference product sets the identity.

Complete square constant, indirect formula shows advantages.

Pay attention to identity

Solve a quadratic equation with one variable

The equation has no linear term, so it is ideal to find the root directly.

If there is no constant term, there is no room for discussion on factorization.

B and c are equal to zero, and the root is also zero. Don't forget.

B and c are not both zero, factorization or formula,

You can also set the formula directly and choose a good prescription according to the topic.

Discrimination of positive proportional function

To judge the proportional function, the test is divided into two steps.

One quantity means another quantity, right?

Look at the numerical values, all you need is real numbers.

Distinguishing the proportional function, the measurement can be divided into two steps.

One quantity means another quantity, right or wrong.

If there is, it depends on the value. All real numbers must be there.

Images and properties of proportional function

The straight line of the proportional function diagram passes through the origin.

K is positive one, three, negative two and four, and the changing trend is in the heart.

K is low on the left and high on the right, climbing in the same direction.

K is negative, left high and right low, one big and one small down the mountain.

linear function

Linear function diagram straight line, passing point.

K is low on the left and high on the right, and the higher you climb, the higher you climb.

K is negative, left high and right low. It's obviously getting lower and lower.

K is called the slope b intercept, and the zero intercept becomes a positive function.

inverse proportion function

Inverse hyperbola, crossing point.

K is plus one, three, minus two, four, and the two axes are its asymptotes.

K is high on the left and low on the right, and one or three quadrants slide down the mountain.

K is negative, low left and high right, and the second and fourth quadrants are like climbing mountains.

quadratic function

Quadratic function appears when the quadratic equation changes from zero to y.

All real numbers define the domain, and images are called parabolas.

A parabola has an axis of symmetry, and both sides are monotonously opposed.

A sets the opening and size, and the intersection of axes is called the vertex.

Vertex is either high or low. Up, down and high are conspicuous.

If you want to draw a parabola, you can also translate the pursuit point.

Extract the formula and set the vertex, and then choose two ways.

After drawing the list, connect the lines and keep the translation rules in mind.

Add parentheses to the left and right, and add and subtract redundant numbers.

When the quadratic equation changes from zero to y, the quadratic function is obtained.

This image is called a parabola and defines all the real numbers in the field.

A set the opening and size, with the opening facing upward.

The absolute value is large and the opening is small, and the opening is negative.

Parabola has an axis of symmetry, and the increase and decrease characteristics can be seen in the figure.

The intersection of axes is called the vertex, and the ordinate of the vertex is the most valuable.

If you want to draw a parabola, trace the point and translate two roads.

Select formulas to fix vertices and draw all translation points.

After drawing points in the list, connect them, and the three points roughly define the whole picture.

If you want to translate, it is not difficult to draw a basic parabola first

The vertex moves to the new position, and the size of the opening follows the foundation.

Parabola of note cardinal number

Lines, rays and line segments

Linear rays are related to line segments and similar shapes.

The length of a straight line is uncertain and can extend to both sides indefinitely.

The ray has only one endpoint and extends in a straight line in the opposite direction.

The two ends of the line segment are fixed in length and extend in two directions to form a straight line.

The alignment of two points is * * *, and forming a graph is the most common.

corner

Starting from a point, two rays form a figure called an angle.

The opposite direction of the * * line is a right angle, and half of the right angle is called a right angle.

A right angle is twice as large as a fillet, and the one smaller than the right angle is called an acute angle.

Straight and peaceful are obtuse angles, and flat and round are called optimal angles.

The sum of two complementary angles and a right angle is the complementary angle of a right angle.

Starting from a point, two rays form a figure called an angle.

A right angle is opposite to a straight line, and half of the right angle is called a right angle.

A right angle is twice as large as a fillet, and the one smaller than the right angle is called an acute angle.

The obtuse angle is between straight and flat, and the perimeter of the flat is called the optimal angle.

Harmony as a right angle is called complementarity, a complementary angle, and a straight angle.

Prove equal product or proportional line segment

Equal product or proportional line segments can be proved in many ways.

The equal product of the certificate should be changed to equal ratio, and the characteristics should be seen according to the graph.

* * * Points intersect with * * lines, and parallel sections prove the problem.

The three stereotypes are very similar. Try to prove the similarity.

The graphics are obviously not similar, and the equal line segment ratio replaces the certificate.

After the exchange, the conclusion can be established and the original proposition is proved.

It is really impossible to use the area, and the projection angle can be divided into lines.

As long as you are willing to climb, you can use your hands and your brain.

Solve irrational number equation

Everything has two sides, and everything has two sides.

There is no trace of the root sign of the power, and the equation can be solved without burden.

Neither is relatively difficult, and it is easy to multiply twice.

Under special circumstances, it is inevitable to exchange elements and get the solution of root test.

Solve fractional equation

Subtract first and then multiply by the common denominator, and the whole equation will be transformed.

Special circumstances can be exchanged, and removing the denominator is the way out.

After obtaining the solution, we should check the root and leave the original to increase the ambiguity.

Solving application problems with column equations

Solve application problems with column equations, and set double solutions for column solutions as far as possible.

There are two ways to establish a direct solution by checking the problem and finding the unknown.

List drawing creates equations and follows rules when solving equations.

The test is accurate, meets the meaning of the question, and the answer is the same.

Add auxiliary line

Learning geometry is a deep experience, and success or failure may be a thread.

Decentralization conditions should be concentrated, and auxiliary lines should often be added.

Don't be afraid and then change your mind.

Practice makes perfect, and insight depends on practice.

It is known that there is a middle line in the figure, and the double-length middle line connects the lines.

The rotating structure is conformal, and the angles of equal line segments can be replaced.

The midline can be obtained by connecting multiple midlines with the midpoint.

If the bisector of an angle is known, the two sides can be perpendicular.

You can also fold along a straight line and stand in an congruent figure.

If you add a vertical line to the bisector, you can see an isosceles triangle.

The angle dividing line and the parallel line change the angular position of the equal line segment.

The vertical line in the line segment is known and connects two equal line segments.

The auxiliary line must be dotted, so it will be linked with the original picture.

Formula of distance between two points

Find the distance between two coaxial points and subtract it greatly.

Two points equidistant from the axis, the same is true for distance calculation.

At any two points on the plane, the horizontal and vertical standard deviations are calculated first.

The difference square plus square, the distance formula should be kept in mind.

Determination of rectangle

Any quadrilateral, three right angles form a rectangle;

Diagonal lines are divided into equal parts, and quadrangles are rectangles.

Known parallelogram, a right angle is called rectangle;

If two diagonal lines are equal, they are naturally rectangles.

Determination of diamond shape

Any quadrilateral, four sides are equal to form a diamond;

The diagonal of the quadrilateral and the perpendicular bisector are diamonds.

It is known that a parallelogram with equal adjacent sides is called a rhombus;

If two diagonal lines are perpendicular, it is a diamond in logic.