Who can help explain the several ways to solve a quadratic equation?

There are four ways to solve a quadratic equation:

1. Direct leveling method; 2. Matching method; 3. Formula method; 4. Factorization method 1. The direct square method is a method of solving a quadratic equation by using the direct square.

2。The matching method is the method of obtaining the root of a one-dimensional quadratic equation by matching the method of matching a perfect flat method.

3。In the formula method, the square difference of two numbers is equal to the product of the sum of the two numbers and the difference between the two numbers. The sum of the squares of the two numbers plus (or subtract) twice the product of the two numbers is equal to the square of the sum (or difference) of the two numbers.

4。A method used in mathematics to solve higher-order unary equations. The method of factoring the number (including unknown numbers) on one side of the equation to 0 by moving it, and turning the other side of the equation into the product of several factors, and then making each factor equal to 0 to find its solution is called factorization.

1] General solution: It is to find a way to decompose the left side of the equation into a factor, so as to achieve the purpose of descending, and find the solution of the equation. a) Formula method:

The formula for the square of the sum and the square formula for the difference, e.g. x 2+2x+4+0, (x+2) 2=0 4x 2-8x+4=0 (2x-2) 2=0, is also used to apply the above formula. b) Cross multiplication method (see the example problem in the book for the solution) 2] Universal solution: substitute the root finding formula.

For problems that cannot be solved by the above methods, the root finding formula is used to solve them.

1. Factoring method: The principle of factorization method is to use the sum of squares formula (a b) 2=a2 2ab+b2 or the square difference formula (a + b) (a-b) = a2-b2, and use the formula upside down. For example, x2+4=0 can be used to use the squared difference formula, and 4 can be regarded as 22, that is, x2+22 =>x-2)(x+2) and then solved separately.

Multiplying 0 by any number gives 0, (x-2) if 0 then x=2, (x+2) equals 0 then x=-2, and that's it.

2. Matching method: The matching method is not very difficult but very important, the matching method can find the vertices and coordinates of the quadratic function, and can also solve the quadratic equation. The first step is to transform it into the form ax2+bx=c.

In the second step, take the square of half of the coefficient b of the primary term, and then equation. b=8, take half first, that is, 4, and then the square is 16, and add both sides at the same time, which is x2+8x+16=2+16. If you change the shape, the sum of squares is reversed, and Zhaoyou 16 is regarded as 42, which is (x+4)2=18.

Family search and then directly open the square, x+4 = 18, and then move the term to simplify, x= 3 2-4. Then write out the solutions separately and you're done.

3. Formula method: The formula method is relatively simple, 2x2-x=6 is first transformed into the form of the general form ax2+bx+c=0, and then find a, b, c, and then directly apply the formula (-b b2-4ac) 2a, δ b2-4ac 0 has two unequal real number missing hidden roots, δ b2-4ac 0 has two equal real number roots, and the solution is x1 2 x2 -2 3

How to solve a quadratic equation

An integer equation that contains two unknowns and the number of terms containing unknowns is 1 is called a binary equation. All binary linear equations can be reduced to the general expression of ax+by+c=0(a, b≠0) and the standard expression of ax+by=c(a, b≠0), otherwise they are not binary linear equations.

The value that fits each pair of unknowns of a binary equation is called a solution of the binary equation. Each binary equation has an infinite number of solutions to the equation, and only the system of binary equations composed of binary equations may have a unique solution, and the system of binary equations is often solved by addition and subtraction or substitution of the elimination method into a unitary equation.

An unknown number of an equation in a system of equations is expressed by an algebraic formula containing another unknown, substituted into another equation, an unknown is eliminated, a unary equation is obtained, and finally the solution of the system of equations is obtained.

The general steps of solving a system of binary linear equations by substitution elimination method are hand open:

1) Equal substitution: select an equation with relatively simple coefficients from the system of equations, and express one unknown number (such as y) in this equation with the algebraic formula of another unknown number (such as x), that is, write the equation in the form of y=ax+b;

2) Substituting elimination: substituting y=ax+b into another equation, subtracting y, and obtaining a one-dimensional equation about x;

3) Solve this unary equation and find the value of x;

4) Substitution: substitute the value of x into y=ax+b to find the value of y, so as to obtain the solution of the system of equations;

The four-dimensional solution of a quadratic equation. 1. Formula method.

Second, the matching method.

3. Direct leveling method.

Fourth, factorization.

Formula 1 first determines =b -4ac, if <0 the original equation has no real root;

2If =0, the original equation has two identical solutions: x=-b (2a);

3 If >0, the solution of the fibrillary permeability equation is: x=((b) 2a).

Matching method. First, move the constant c to the right of the equation to get ax +bx=-c.

The quadratic term coefficient is reduced to 1 to obtain: x + (b a) x = - c a, and the square of half of (b a) on both sides of the equation is added to x +(b a) x+(b (2a)) c a+(b (2a)) The equation is: (b + (2 a)) c a + (b (2a)).

5. If -c a+(b (2a)) 0, the original equation has no real root;

If -c a+(b (2a)) 0, the original equation has two identical destructive ridges and is solved as x=-b (2a);

If -c a+(b (2a)chenna) >0, the solution of the original equation is x=(-b) b -4ac)) 2a).

The three basic methods for solving a quadratic equation are as follows:

1. Direct leveling method, this method is used in simple solution equations, but it should be noted that the coefficient of the quadratic term should be turned into "1" before doing it.

2. Allocation method. This method is used very frequently, and it can be used in basically simple problems of solving one-dimensional quadratic equations, and it is also very fast. Note that the coefficient of the quadratic term is first converted to "1", and then arranged into a perfect square method, so that you can use the method of perfect square formula in the factorization you learned before to solve the problem.

3. Formula method; The formula method is commonly known as the universal method, which can be used for any problem of solving a one-dimensional quadratic equation; However, the formula needs to be memorized, and the amount of solution is large when doing the problem, so it is not recommended. When doing the problem, you can observe it first, and if the method you learned before is not applicable, then use the formula method.

4. The common factor method in the factorization method. Factorization is a very important method to solve problems, and factorization, as an important chapter of the eighth grade, plays an important role in the calculation problems.

Therefore, factorization must be learned, and there are many knowledge points in the design, which requires a review of the knowledge learned before. The common factor rule is the basic method in factorization, as long as the common factor is proposed, the solution of the equation is much simpler.

General solution.

1.Matching method.

solves all one-dimensional quadratic equations).

For example, solve the equation: x 2 + 2 x 3 = 0

Solution: Shift the constant term to obtain: x 2 + 2x = 3

Add 1 to both sides of the equation at the same time (to form a perfect square) to obtain: x 2 + 2x + 1 = 4

Factoring yields: (x+1) 2=4

Solution: x1=-3, x2=1

Use the matching method to solve the small formula of the one-dimensional quadratic equation.

The quadratic coefficient is one.

The constants should be shifted to the right.

The coefficient is half square at a time.

Both sides add the most quite.

2.Formula method.

solves all one-dimensional quadratic equations).

First of all, we need to determine how many roots a quadratic equation has by using the discriminant expression of the root of δ=b 2-4ac.

1.When δ=b 2-4ac<0 x no real root (junior).

2.When δ=b 2-4ac=0, x has two identical real roots, i.e., x1=x2

3.When δ=b 2-4ac>0, x has two different real roots.

When the judgment is completed, if the equation has a root, the root belongs to the two cases, and the equation has a root, then the formula can be used: x= 2a

to find the root of the equation.

3.Factorization.

Partially solvable one-dimensional quadratic equations) (the factorization method is divided into the "common factor method", the "formula method" (and the "square difference formula" and the "perfect square formula") and the "cross multiplication method".

For example, solve the equation: x 2 + 2 x + 1 = 0

Solution: Factoring by the perfect square formula: (x+1 2=0.)

Solution: x1=x2=-1

4.Direct leveling method.

Partial unary quadratic equations can be solved).

5.Algebraic method.

solves all one-dimensional quadratic equations).

ax^2+bx+c=0

Divide by a at the same time, which becomes x 2 + bx a + c a = 0

Let x=y-b 2

The equation becomes: (y 2+b 2 4-by)+(by+b 2 2)+c=0 x error should be (y 2+b 2 4-by) divided by (by-b 2 2)+c=0

Then become: y 2+(b 22*3) 4+c=0 x y 2-b 2 4+c=0

y=±√[b^2*3)/4+c] x __y=±√[b^2)/4+c]

Methods of elimination:

Substituting the elimination method, (commonly used) addition and subtraction elimination method, (commonly used) sequential elimination method, (this method is not commonly used) the order is correct.

Examples of the elimination method in this paragraph:

x-y=3 3x-8y=4 is substituted by x=y+3 to get 3(y+3)-8y=4 y=1 so x=4 then: the solution of this system of binary equations x=4 y=1

There are several solutions that are not in this textbook, but are more applicable:

a) Addition, subtraction, substitution and mixed use methods. Example 1, 13x+14y=41 (1) 14x+13y=40 (2) Solution: (2)-(1) to get x-y=-1 x=y-1 (3) Substitute (3) into (1) to get 13(y-1)+14y=41 13y-13+14y=41 27y=54 y=2 Substituting y=2 into (3) to get x=1 So:

x=1, y=2 Finally x=1 , y=2, solution Features: Add and subtract two equations, a single x or a single y, so that the next substitution elimination element is applied. (2) The substitution method is another method of binary equations, that is, bringing one equation into another equation such as:

x+y=590 y+20=90%x is substituted as: x+90%x-20=590 Example 2: (x+5)+(y-4)=8 (x+5)-(y-4)=4 Let x+5=m,y-4=n The original equation can be written as m+n=8 m-n=4 The solution gives m=6,n=2 So x+5=6,y-4=2 So x=1,y=6 Features:

Both equations contain the same algebraic formula, such as x+5, y-4 and so on, and the main reason is that the equation can be simplified after the change. (3) Alternative exchange Example 3, x:y=1:

4 5x+6y=29 Let x=t, y=4t Equation 2 can be written as: 5t+24t=29 29t=29 t=1 So x=1, y=4

Equations of the shape ax +bx+c=0 are generally solved by factorization, flattening, formula, etc. Please refer to the tutorial for details. Only the formula method is introduced here, i.e., x=[-b (b -4ac)] 2a.