Example: Solve the equation (3x+ 1)? =7
∫(3x+ 1)？ =7
∴3x+ 1= √7
∴x= ﹙﹣ 1 √7﹚/3
The solution of the original equation is x 1 = √ 7 √ 1 √/3, and x2 = √ 7- 1 √/3.
2. Matching method: use matching method to solve equation ax? +bx+c=0 (a≠0)。 First, move the constant c to the right of the equation: ax? +bx=-c, convert the quadratic coefficient into: x? +bx/a=- c/a, the square of half the coefficient of the first term is added to both sides of the equation: x? +bx/a+( b/2a)？ =- c/a+( b/2a)？ How does the left side of the equation become completely flat: (x+b/2a)? = -c/a﹢﹙b/2a﹚？ When b? -4ac≥0，x+b/2a = √-c/a-b/2a？ , so x =-b [√-b? -4ac]-/2a (this is the root formula)
Example: Solving Equation 3x by Matching Method? -4x-2=0
Move the constant term to the right of Equation 3x? -4x=2
Convert the quadratic coefficient into: x? -﹙4/3﹚x= 2/3
On both sides of the equation, add half the square of the first coefficient: x? -﹙4/3﹚x+( 4/6)？ =2/3 +(4/6 )?
Formula: (x-4/6)? = 2/3 +(4/6 )?
Direct square: x-4/6 = √ [2/3+(4/6)? ]
∴x= 4/6 √[2/3 +(4/6)？ ]
The solutions of the original equation are x1= 4/6 √10/9 √, x2 = 4/6 √19 √.
3. Formula method: convert the quadratic equation of one variable into a general form, and then calculate the discriminant △=b? The value of -4ac, when b? When -4ac≥0, substitute the values of the coefficients a, b and c into the root formula X = [-B √ (B? -4ac)]/(2a)，(b？ -4ac≥0) can get the root of the equation.
Example: Solving Equation 2x by Formula Method? +4x+ 1=0
∴a=2,b=4，c= 1
⊿=b？ -4ac = 16-4 * 2 * 1 = 8 & gt； 0
x =(-b √⊿)/(2a)=(-4 2√2)/4 =(-2√2)/4
The solution of the original equation is x1= (-2+√ 2)/4x2 = = (-2-√ 2)/4.
4. Factorial decomposition method: the quadratic trinomial on one side of the equation is decomposed into the product of two linear factors, so that the two linear factors are equal to zero respectively, and two linear equations are obtained. The roots obtained by solving these two linear equations are the two roots of the original equation. This method of solving linear quadratic equations is called factorization.
Example: factorization to solve the equation: 6x? +5x-50=0
6x？ +5x-50=0
(2x-5)(3x+ 10)=0
2x-5 = 0 or 3x+ 10=0.
∴ The solution of the original equation x 1=5/2, x2=- 10/3.
Summary:
Usually, factorization is the most commonly used method to solve quadratic equations with one variable. When factorization is applied, the equation is written in a general form and the quadratic coefficient is turned into a positive number.
Direct leveling method is the most basic method.
Formula and collocation are the most important methods.
Formula method is suitable for any quadratic equation with one variable (some people call it universal method). When using the formula method, the original equation must be transformed into a general form to determine the coefficient, and before using the formula, the value of the discriminant should be calculated to judge whether the equation has a solution.
Matching method is a tool to derive formulas. After mastering the formula method, you can directly use the formula method to solve the quadratic equation of one variable, so you generally don't need to use the matching method to solve the quadratic equation of one variable. However, matching method is widely used in the study of other mathematical knowledge, and it is one of the three important mathematical methods (method of substitution, matching method and undetermined coefficient method) required to be mastered in junior high school, so you must master it well.