First, the main points of knowledge:

One-dimensional quadratic equation and one-dimensional linear equation are both integral equations, which are a key content of junior high school mathematics and the basis of studying mathematics in the future.

The foundation should attract students' attention.

The general form of the unary quadratic equation is: ax2 bx c=0, (a≠0), which contains only one unknown, and the highest order of the unknown is 2.

The whole equation of.

The basic idea of solving quadratic equations with one variable is to simplify them into two quadratic equations with one variable. Quadratic equation with one variable has four solutions.

Methods: 1, direct Kaiping method; 2. Matching method; 3. Formula method; 4. Factorial decomposition method.

Second, detailed methods and examples:

1, direct Kaiping method:

The direct Kaiping method is a method to solve a quadratic equation with a direct square root. Solving (x-m)2=n (n≥0) by direct Kaiping method.

The solution is an equation of x = m.

Example 1. Solve the equation (1) (3x1) 2 = 7 (2) 9x2-24x16 =1.

Analysis: (1) This equation is obviously easy to do by direct flattening, (2) The left side of the equation is completely flat (3x-4)2, and the right side =11>; 0, so

This equation can also be solved by direct Kaiping method.

(1) solution: (3x 1)2=7×

∴(3x 1)2=5

∴ 3x 1 = (be careful not to lose the solution)

∴x=

The solution of the original equation is x 1=, x2=.

(2) Solution: 9X2-24x16 =11.

∴(3x-4)2= 1 1

∴3x-4=

∴x=

The solution of the original equation is x 1=, x2=.

2. Matching method: use matching method to solve equation ax2 bx c=0 (a≠0).

First, move the constant c to the right of the equation: ax2bx =-c.

Convert the quadratic term into 1: X2x =-

Add half the square of the coefficient of the first term on both sides of the equation: x2 x ()2=- ()2.

The left side of the equation becomes completely flat: (x )2=

When b2-4ac≥0, x =

∴x= (this is the root formula)

Example 2. Solving Equation 3x2-4x-2=0 by Matching Method

Solution: Move the constant term to the right of equation 3x2-4x=2.

Transform the quadratic term into 1: x2-x =

Add half the square of the coefficient of the first order term on both sides of the equation: x2-x ()2= ()2.

Formula: (x-)2=

Direct square: x-=

∴x=

The solution of the original equation is x 1=, x2=.

3. Formula method: convert the quadratic equation of one variable into a general form, and then calculate the value of the discriminant △=b2-4ac. B2-4ac≥0, release all items.

Substitute the values of coefficients A, B and C into the formula x=(b2-4ac≥0) to get the root of the equation.

Enthusiastic users | 20 13- 10-27

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