At the same time, we should fully realize that learning quadratic equation well can lay a solid foundation for learning quadratic inequality, exponential equation, logarithmic equation, trigonometric equation, function, quadratic curve and so on.
Quadratic function is the most direct example. The unary quadratic equation ax2+bx+c=0(a0) is a special case of quadratic function y=ax2+bx+c(a0) when y=0. To learn quadratic equation well, we must first learn these basic knowledge, such as the basic operations of real numbers and algebraic expressions, quadratic equation with one variable and so on.
What is a quadratic equation with one variable?
An integral equation with an unknown number and the highest degree of the unknown number is 2 is called a quadratic equation.
The unary quadratic equation ax2+bx+c=0(a0), which is characterized in that the left side of the equation is a quadratic polynomial about the unknown quantity X, and the right side of the equation is zero, where ax2 is called a quadratic term and A is called a quadratic term coefficient; Bx is called a linear term, and b is called a linear term coefficient; C is called a constant term.
Mathematics of senior high school entrance examination, quadratic equation of one variable, analysis of typical examples 1;
Do you know the quadratic equation x2 of x? 6x+(2m+ 1)=0 has a real root.
(1) Find the range of m;
(2) If the two real roots of the equation are x 1, x2, 2x 1x2+x 1+x220, find the range of m 。
Solution: (1) According to the meaning of the question, = (? 6)2? 4(2m+ 1)0，
Solve for M4;
(2) According to the meaning of the question, x 1+x2=6, x 1x2=2m+ 1,
And 2x 1x2+x 1+x220,
So 2(2m+ 1)+620，m3，
And m4,
So the range of m is 3m4. ..
Stem analysis:
(1) According to the meaning of the discriminant, we get = (? 6)2? 4(2m+ 1)0, and then solve the inequality; (2) According to the relationship between root and coefficient, we can get x 1+x2=6, x 1x2=2m+ 1, and then we can get 2 (2m+1kloc-0/)
Thinking about solving problems:
This question examines the relationship between roots and coefficients: If x 1 and x2 are two roots of the unary quadratic equation ax2+bx+c=0(a0), x 1+x2=? B/a, x 1x2=c/a, and the relationship between root and coefficient is also investigated.
Memorize the solution of a quadratic equation;
1, direct Kaiping method
Using the definition of square root to find the solution of quadratic equation in one variable is called direct Kaiping method. The direct Kaiping method is suitable for solving the quadratic equation with one variable in the form of (x+a) 2 = b.
2. Matching method
Matching method is an important mathematical method, which is not only suitable for solving quadratic equations with one variable, but also widely used in other fields of mathematics. The theoretical basis of the matching method is the complete square formula A2A2 AB+B2 = (AB) 2. If A in the formula is regarded as an unknown X and replaced by X, there will be x22xb+b2=(xb)2.
3. Formula method
Formula method is a method to solve the quadratic equation of one variable by finding the root formula, and it is a general method to solve the quadratic equation of one variable.
4, factorization method
Factorization is to find the solution of the equation by factorization. This method is simple and easy to use, and it is the most commonly used method to solve the quadratic equation of one variable.
The four schemes have their own characteristics, and the basic idea is to reduce the order. Only by grasping them accurately can we easily solve the equations. It is worth noting that although the formula method is omnipotent and applicable to any quadratic equation with one variable, it is not necessarily the simplest. Therefore, when solving equations, we should first consider whether simple methods such as "direct Kaiping method" and "factorization method" can be applied. If not, we should consider the formula method (if appropriate, we can also consider the collocation method) When there are brackets in the equation, we should first consider whether there is a simple method with overall thinking. If we can't find the right way,
Mathematics of senior high school entrance examination, quadratic equation of one variable, analysis of typical examples II;
The univariate quadratic equation x2 about x is known. (2m+3)x+m2+2=0。
(1) If the equation has real roots, find the range of the number m;
(2) If the two real roots of the equation are x 1 and x2, respectively, and x12+x22 = 31+x1x2, find the value of the number m. 。
Solution: (1) A quadratic equation x2 about x? (2m+3)x+m2+2=0 has real roots,
0, that is, (2m+3)2? 4(m2+2)0，
m？ 1/ 12;
(2) According to the meaning of the question, x 1+x2=2m+3, x 1x2=m2+2,
x 12+x22 = 3 1+x 1x 2，
(x 1+x2)2？ 2x 1x 2 = 3 1+x 1x 2，
That is (2m+3)2? 2(m2+2)=3 1+m2+2，
The solution is m=2, m=? 14 (discarded),
m=2。
Test center analysis:
Discrimination formula of roots; The relationship between roots and coefficients.
Stem analysis:
(1) According to the meaning of the discriminant of roots, we get 0, that is, (2m+3)2? 4(m2+2)0, which exactly solves the inequality;
(2) According to the relationship between root and coefficient, x 1+x2=2m+3, x 1x2=m2+2, and (x 1+x2)2 is obtained under the condition of known re-deformation. 4x1x 2 = 31+x1x 2, and the result can be obtained by substitution.
Thinking about solving problems:
This question examines the discriminant of the root of the unary quadratic equation ax2+bx+c=0(a0) =b2. 4ac: When > 0, the equation has two unequal real roots; When =0, the equation has two equal real roots; When < 0, the equation has no real root.
This problem also examines the relationship between the roots and coefficients of a quadratic equation.
The relationship between roots and coefficients of quadratic equation in one variable;
If the two real roots of the equation ax2+bx+c=0(a0) are x 1, x2, then x 1+x2=? B/a, x1x2 = c/a. That is to say, for any real root of a quadratic equation, the sum of the two roots is equal to the reciprocal of the quotient obtained by dividing the coefficient of the first term of the equation by the coefficient of the second term; The product of two roots is equal to the quotient obtained by dividing the constant term by the coefficient of quadratic term.
It is worth noting that b2-4ac? 0.
In the unary quadratic equation ax2+bx+c=0(a0), b2? 4ac is called the discriminant of the root of unary quadratic equation ax2+bx+c=0(a0), which is usually represented by "",that is, =b2? 4ac。
When > 0, the equation has two unequal real roots;
When =0, the equation has two equal real roots;
When < 0, the equation has no real root.
By solving the quadratic equation of one variable and using it to solve practical problems, we should learn to use mathematical thinking methods such as transformation in the process of solving these problems.
Frankly speaking, it is a prerequisite to learn the related concepts and solutions of quadratic equations in one variable. If we want to extract a quadratic equation in real life and use it to solve practical problems, then we must learn how to change our thinking.
Mathematics for the senior high school entrance examination, quadratic equation with one variable, analysis of typical examples 3:
In 20 17, a certain place allocated12.8 million yuan for resettlement in different places, and plans to increase investment year by year. 20 19 Increase the investment by160,000 yuan on the basis of 20 17.
(1) From 20 17 to 20 19, what is the average annual growth rate of immigrant funds invested in this area?
(2) In the concrete implementation of relocation in 2019, the district plans to invest no less than 5 million yuan for priority relocation and rental incentives, and 8 yuan will be rewarded every day before 1000 households (including 1000 households), and 5 yuan will be subsidized every day after 1000 households for 400 days.
Solution: (1) suppose that the average annual growth rate of immigrant funds invested in this area is X. According to the meaning of the question,
de: 1280( 1+x)2 = 1280+ 1600，
Solution: x=0.5 or x= 2.25 (house),
A: From 20 17 to 20 19, the average annual growth rate of immigrant funds invested in this area was 50%;
(2) Suppose that a resident in this district enjoys the priority to move the rent this year. According to the meaning of the question,
Get: 10008400+(a? 1000)54005000000,
Solution: a 1900,
A: This year, at least 65,438+0,900 families in this area have enjoyed preferential relocation and rental incentives.
Test center analysis:
Application of quadratic equation in one variable.
Stem analysis:
(1) Let the average annual growth rate be X, and list the available equations according to the following: 20 17-year investment (1+ growth rate) 2 = 20 19-year investment;
(2) Suppose that a resident in this district enjoys the priority of moving and renting this year. According to the total number of rewards won by the first 1000 households+1000 households, the total number of rewards won after that is 5 million, and the inequality can be solved.
Mathematics of senior high school entrance examination, quadratic equation of one variable, analysis of typical examples 4:
Qinghai News Network News: On February 2, 2065438, the first green road free bicycle rental system in Xining was officially put into use. The municipal government invested1120,000 yuan this year to build 40 bicycle stations with 720 bicycles. In the future, the investment will increase year by year.
(1) What is the cost and unit price of public bicycles per station?
(2) Please calculate the average annual growth rate of bicycles allocated by the municipal government from 20 16 to 20 18.
Test center analysis:
Application of quadratic equation in one variable: application of linear equation in two variables.
Stem analysis:
(1) Invest1120,000 yuan respectively to build 40 public bicycle stations and allocate 720 public bicycles, and invest 3.405 million yuan to build120 public bicycle stations and allocate 2,205 public bicycles.
(2) Using 720 bicycles deployed in 20 16 years, combined with the growth rate of X, the number of bicycles deployed in 20 18 years is obtained, and the equation is solved to get the answer.
With the deepening of the new curriculum reform, the current senior high school entrance examination more and more examines the comprehensive ability of candidates, such as applying mathematical knowledge to solve specific problems. In the usual learning process, we should combine the knowledge structure and specific problems of the quadratic equation of one variable, list the knowledge network diagram, actively explore and find problems, ask questions from special to general, constantly improve our thinking ability, optimize our learning methods, master the corresponding problem-solving methods, and learn the quadratic equation of one variable with more hands, brains and mouths.