An integral equation with only one unknown number and the highest degree of the unknown number is 2 is called a quadratic equation with one variable.
The quadratic equation with one variable has three characteristics: (1) contains only one unknown; (2) The maximum number of unknowns is 2; (3) It is an integral equation. To judge whether an equation is an unary quadratic equation, we should first look at whether it is an integral equation. If so, then tidy it up. If it can be arranged in the form of AX 2+BX+C = 0 (A ≠ 0), then this equation is an unary quadratic equation.
general formula
Ax 2+bx+c = 0 (a, b, c are constants, a≠0).
For example: x2- 1 = 0
general solution
1. direct Kaiping method
2. Matching method
3. Formula method
4. Factorization method
Identification method
The judging formula of quadratic equation in one variable: b 2-4ac
b^2-4ac>； The 0 equation has two unequal real roots.
Equation B 2-4ac = 0 has two equal real roots.
b^2-4ac<； Equation 0 has no real root.
The above can be pushed from left to right and vice versa.
Steps to solve a quadratic equation with one variable
(1) Analyze the meaning of the problem and find out the equivalent relationship between the unknown in the problem and the conditions given in the problem;
(2) Set unknowns, using algebraic expressions of set unknowns to represent other unknowns;
(3) Find out the equation relation and use it to list the equations;
(4) Solving the equation to find the value of the unknown quantity in the problem;
(5) Check whether the required answer meets the meaning of the question and make an answer.
Problem solving thinking
1. Change your mind 0
Transforming thinking is the most common way of thinking in junior high school mathematics.
With the idea of transformation, unknown problems can be transformed into known problems, and complex problems can be transformed into simple problems. In this chapter, the solution of a quadratic equation is transformed into a square root problem, and the quadratic equation is transformed into a linear equation through factorization.
2. Ideas from special to general
From the special to the general is the universal law of our understanding of the world. Through the study of special phenomena, we can draw general conclusions, such as using direct Kaiping method to solve special problems, using formula matching method to explore the relationship between roots and coefficients of quadratic equations in one variable.
3. The idea of classified discussion
The discriminant of the roots of a quadratic equation in one variable embodies the idea of classified discussion.
4. Substitution method: set an algebraic expression or fraction in the equation as a letter and substitute it for calculation. The process is simple.
Elaboration of classic examples.
1. For the definition of a univariate quadratic equation, we should fully consider the three characteristics of the definition, and don't ignore that the quadratic term coefficient is not 0.
2. When solving a quadratic equation with one variable, according to the characteristics of the equation, choose the solution method flexibly, first consider whether the direct Kaiping method and factorization method can be used, and then consider the formula method.
3. There are both positive and negative discriminant of the root of the unary quadratic equation (a≠0), which can be used to solve the equation and determine the root of the equation (1); (2) Determine the range of roots according to the properties of parameter coefficients; (3) Solve the problem of proof related to roots.
4. The roots and coefficients of a quadratic equation with one variable have many applications: (1) Knowing one root of the equation, finding another root and parameter coefficients without solving the equation; (2) Knowing the equation, finding the value of algebraic expression with two symmetrical expressions and related unknown coefficients; (3) Given two equations, find the root of an unary quadratic equation.
Vieta theorem
Vieta's Theorem: In the unary quadratic equation AX 2+BX+C (A is not 0).
Let two roots be X 1 and X2.
Then x1+x2 =-b/a.
X 1*X2=c/a
Vieta's theorem can also be used for higher-order equations. Generally speaking, for an equation with n ∑ AIX I = 0.
Its roots are expressed as X 1, X2…, Xn.
we have
∑xi=(- 1)^ 1*a(n- 1)/a(n)
∑XiXj=(- 1)^2*A(n-2)/A(n)
…
∏Xi=(- 1)^n*A(0)/A(n)
Where ∑ is the sum and ∏ is the quadrature.
Pay special attention to
When the quadratic equation of one variable cannot be solved at one time, it must be changed into a general form first, and then the square root must be found. The final number must be two or none.