The ninth grade mathematics mind map: ninth grade mathematics appreciation: sorting out the knowledge points of unary quadratic equation I. Definition and characteristics
1, unary quadratic equation: An integral equation with an unknown number whose highest degree is 2 is called unary quadratic equation.
2. The general form of the unary quadratic equation: ax +bx+c=0 squared (a? 0), which is characterized in that a quadratic polynomial about the unknown quantity X is added to the left side of the equation, and the right side of the equation is zero, where the square+of ax is called quadratic term, and A is called quadratic term coefficient; Bx is called a linear term, and b is called a linear term coefficient; C is called a constant term.
Second, the origin of the equation
The pottery pieces left by ancient Babylon show that in about 2000 BC, the mathematicians of ancient Babylon were able to solve the quadratic equation of one variable. In about 480 BC, China people had found the positive root of quadratic equation by collocation method, but did not put forward the general solution. Around 300 BC, Euclid proposed a more abstract geometric method to solve quadratic equations.
Brahmagupta of India in the 7th century was the first person who knew how to use algebraic equations, which allowed positive and negative roots.
In 1 1 century, Elazemi of Arabia independently developed a set of formulas for finding positive solutions of equations. Abraham? Bachillat (also known as Sawasoda in Latin) introduced the complete solution of the quadratic equation of one variable to Europe for the first time in his book Liber embadorum.
It is said that Schridde Haller was one of the first mathematicians to give the general solution of quadratic equation. But this was controversial in his time. The rule to solve this problem is (quoted from Pashgaro II):
Both sides of the equation are multiplied by four times the coefficient of the unknown quadratic term at the same time;
At the same time, the square of the coefficient of the unknown term is added to both sides of the equation;
Both sides of the equation are opened twice at the same time.
Third, nature.
The two roots of the equation have the following relations with the numbers in the equation: x 1+x2= -b/a, x 1? X2=c/a (also known as Vieta's theorem)
When two of the equations are x 1 and x2, the equations are: x 2+(x1+x2) x+x1x 2 = 0 (derived from Vieta theorem).
b^2-4ac>； 0 has two unequal real roots, and b 2-4ac = 0 has two equal real roots, and b 2-4ac.
Fourth, the general solution
The univariate quadratic equation has the following general solution:
Matching method (solvable partial quadratic equation)
Formula method (in junior high school, one-dimensional quadratic equations can be solved, provided that: △? 0)
Factorization method (which can solve some quadratic equations with one variable)
Direct Kaiping method (which can solve all quadratic equations in one variable)
Ninth grade mathematics: the basic solution of a quadratic equation: the basic solution of a quadratic equation? Demote? The quadratic equation of one variable is transformed into a linear equation of one variable to solve.
1. direct Kaiping method: for (x+a)2 =b(b? 0) Square the two sides directly and convert it into two linear equations.
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.
2. Matching method: use matching method to solve the quadratic equation of one variable: ax2 +bx+c=0(k? 0) The general steps are:
(1) into a general form;
② Shift the term, that is, move the constant term to the right of the equation;
③ The coefficient of quadratic term is 1, that is, the coefficient of two sides of the equation divided by quadratic term;
(4) Formula, that is, the square of two sides of the equation plus half of the coefficient of the first term; Transform the original equation into the form of (x+a)2 =b;
(5) if b? 0, the solution of the equation can be obtained by the square roots of both sides; If b? 0, the original equation has no solution.
Basis: The theoretical basis of the matching method is the complete square formula A? 2; +b？ 2; ? 2ab=(a？ b)？ 2;
Key: The key of the matching method is to first change the quadratic coefficient of the unary quadratic equation into 1, and then add half the square of the quadratic coefficient on both sides of the equation.
3. Formula method: Formula method is a method to find the solution of quadratic equation with one variable by using root formula. It is derived from the formula. The root formula of unary quadratic equation is
(b2 -4ac？ 0)。 Steps:
① Convert the equation into a general form;
② Determine the values of A, B and C;
③ Find the value of b2 -4ac, when b2 -4ac? 0 era rooting formula.
4. Factorization: The method of finding the root of a quadratic equation with one variable by factorization is called factorization. Theoretical basis: If ab=0, then a=0 or b=0.
These steps are:
① Turn the right side of the equation into 0;
② decompose the left side of the equation into the product of two linear factors;
(3) Make each factor equal to 0 and get two unary linear equations. Solve these two linear equations with one yuan, and their solutions are all the solutions of the original quadratic equation with one yuan.
Factorization methods: common factor, formula and cross multiplication.
5. Mirror image solution: The geometric meaning of the root of the quadratic equation is the X coordinate of the intersection of the mirror image (a parabola) of the quadratic function and the X axis.