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An integral equation with only one unknown number and the highest exponential power of the unknown number is 2 is called a quadratic equation.

There are four solutions to the quadratic equation of one variable, namely, direct Kaiping method, collocation method, formula method and factorization method. Formula method can solve all quadratic equations of a variable, but formula method can't solve equations without real roots (that is, b? -4ac & lt； 0 equation). Factorization must turn the right side of the equal sign into 0. The matching method is relatively simple: first, the quadratic coefficient of the equation is converted into 1, then the constant term is moved to the right of the equal sign, and finally the square of half the absolute value of the quadratic coefficient is added to both sides of the equal sign.

Chinese name: One-variable quadratic equation

Mbth: a quadratic equation with one variable

Type: integral equation

Standard form: ax? +bx+c=0(a≠0)

Root formula: x = [-b √ (b? -4ac)]/2a

Solution: collocation method, formula method and factorization method.

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meet a condition

The quadratic equation of one variable must satisfy three conditions at the same time:

(1) is the whole equation, that is, both sides of the equal sign are algebraic expressions, if there is a denominator in the equation; And the unknown is on the denominator, then this equation is a fractional equation, not a quadratic equation, please pay attention to this!

(2) contains only one unknown number;

③ The maximum number of unknowns is 2.

Equation form

general formula

Generally speaking, any univariate quadratic equation about X can be transformed into the form of ax2+bx+c=0 (a≠0, A, B and C are constants). This form is called the general form of quadratic equation with one variable. The linear term coefficient b and the constant term c can be any real numbers, while the quadratic term coefficient a must be a real number not equal to 0. In order to determine the coefficient of quadratic term first, and then determine the coefficient of linear term and constant term, we must first change the quadratic equation of one variable into a general form.

Variant form

(a and b are real numbers, a ≠ 0);

(a and c are real numbers, a ≠ 0);

(a is a real number, a≠0).

Note: a≠0 is a very important condition.

matched pattern

Double root type

solution method

Kaiping method

The quadratic equation in the form of x2=p or (nx+m)2=p(p≥0) can be solved by direct Kaiping method.

If the equation is in the form, then it can be obtained.

If the equation can be transformed into the form of (p≥0), then the root of the equation can be found.

note:

① The left side of the equal sign is the square of a number, and the right side of the equal sign is non-negative.

② The essence of order reduction is to transform a quadratic equation with one variable into two linear equations with one variable.

The method is to remove the square root according to the meaning of the square root.

Method of completing a square

step

The quadratic equation of one variable is matched into the form of (x+m)2=n, and then it is solved by direct Kaiping method. This method of solving a quadratic equation with one variable is called collocation method.

Steps of solving a quadratic equation with one variable by collocation method;

(1) transforms the original equation into a general form;

② Divide both sides of the equation by the quadratic term coefficient, so that the quadratic term coefficient is 1, and move the constant term to the right of the equation;

③ Add half the square of the coefficient of the first term on both sides of the equation;

④ The left side is matched into a completely flat mode, and the right side is matched into a constant;

⑤ If the right side is non-negative, its solution can be further obtained by direct Kaiping method; If the right side is negative, it is judged that the equation has no real number solution.

The theoretical basis of the matching method is the complete square formula A? +b？ 2ab=(a b)？

The key point of collocation method is: first, convert the quadratic coefficient of a quadratic equation with one variable into 1, and then add the square of half of the quadratic coefficient on both sides of the equation.

for instance

Example 1: Solving Equation 3x2-4x-2 = 0 by Matching Method

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

Convert quadratic term to 1:

Add half the square of the coefficient of the first term to both sides of the equation:

Formula:

Direct square:

∴ , .

The solution of the original equation is.

Root formula method

step

The method of solving quadratic equation with root formula is called root formula method.

The general steps of solving a quadratic equation with one variable by finding the root formula are as follows:

① Turn the equation into a general form and determine the values of A, B and C (pay attention to the symbols);

(2) Find the value of discriminant and judge the condition of root;

③ On the premise of, substitute the values of A, B and C into the formula to calculate and find the root of the equation.

deductive procedure

The derivation process of the root formula of a quadratic equation with one variable is as follows:

(recipes, both sides)

(Simplified).

The formula for finding the root of a quadratic equation with one variable is suitable for the region where the coefficient of the equation is rational, real, complex or arbitrary.

Discriminant in unary quadratic equation: b under radical sign? -4ac

Any two numbers are multiplied by themselves, if any. In some numeric fields, some values have no square root.

Derivation process 2

The derivation process of the root formula of a quadratic equation with one variable is as follows:

The range of a is arbitrary, and the range of c is arbitrary, and b = (a+1) √ C. From the value of a b c, more than1100 million equations can be obtained, which is consistent with factorization.

Verified by the Vedic Law:

Factorization method

Factorization is a method of solving equations by factorization.

Factorization is to first change the right side of the equation into 0, and then change the left side into the product of two linear factors through factorization, so the values of these two factors are likely to be 0, so that the solutions of the two linear equations can be obtained, thus obtaining the original.

graphical method

Equation simplification transforms the solution of a quadratic equation with one variable into the solution of a linear equation with one variable (mathematical simplification idea).

General steps to solve factorization of quadratic equation in one variable;

① Move the term so that the right side of the equation is zero;

② decompose the left side of the equation into the product of two linear factors;

(3) respectively making each factor zero to obtain two unary linear equations;

(4) Solve these two linear equations, and their solutions are the solutions of the original equation.