To put it simply, the method of cross multiplication is that the multiplication on the left side of the cross is equal to the quadratic term coefficient, the multiplication on the right side is equal to the constant term, and the cross multiplication is equal to the linear term coefficient. Cross multiplication can decompose some quadratic trinomials. The key of this method is to decompose quadratic coefficient A into two cross multiplications.

The product a2 of a factor a 1 decomposes the constant term C into the product of two factors c 1, c2 C 1 multiplied by c2, so that a 1.a2+A2c 1 is just a linear term B, and the result can be directly written as: AX2+BX. When the first coefficient is not 1, it often needs to be tested many times, so be sure to pay attention to the sign of each coefficient. Basic formula: x 2+(p+q) χ+PQ = (χ+p) (χ+q) The so-called multiplication is factorization by using the inverse operation of the multiplication formula (x+a) (x+b) = x 2+(a+b) x+ab. For example, divide by X. Therefore, the above formula can be decomposed into: x 2+7x+ 12 = (x+3) (x+4). Another example is the decomposition factor: a 2+2a-15, and the constant of the above formula-15 can be decomposed into 5×(-3). Therefore, a 2+2a-15 = (a+5) (a-3). Explanation: X 2-3x+2 = as follows: x-1 ╳ x -2 left x times X = X 2 right-1 times-

Example 1

Factorization 2x 2-7x+3. Analysis: First, the quadratic term coefficient is decomposed and written in the upper left corner and the lower left corner of the intersection line, then the constant term is decomposed and written in the upper right corner and the lower right corner of the intersection line, and then the algebraic sum is obtained by cross multiplication to make it equal to the linear term coefficient. Decomposition of quadratic coefficient (positive factor only): 2 = 1×. Decomposition constant term: 3 =1× 3 = 3×1= (-3 )× (-1)× (-3) Draw a cross line to indicate the following four situations:11╳ 23. × 65438+2× (-1) =-51-3 ╳ 2-1× (-1)+2× (-3) =-7 After observation, the fourth situation is correct. Then the constant term c can be decomposed and added to get a 1c2+a2c 1. If it is exactly equal to the first-order coefficient of the quadratic trinomial AX2+BX+C, then the quadratic trinomial can be decomposed into the product of two factors a 1x+c 1 and a2x+c2. In this way, the coefficient can be decomposed by drawing a cross line, which helps us to separate the two.

Example 2

Coefficient 6x 2-7x 5. Analysis: According to the method of example 1, decompose the quadratic term coefficient 6 and the constant term -5, and arrange them respectively. There are eight different arrangements, one of which is 2 1 ╳ 3-52× (-5)+3× 65438. Therefore, the original polynomial can be factorized by cross multiplication. The solution 6x 2-7x-5 = (2x+ 1) (3x-5) points out that it often takes many observations to decompose a quadratic trinomial whose quadratic coefficient is not 1 by cross multiplication. We can determine whether we can use cross multiplication to decompose factors. For the quadratic trinomial with quadratic coefficient 1, we can also use cross multiplication to decompose the factors. At this time, we only need to consider how to decompose the constant term. For example, if we factorize X 2+2x- 15, then the cross multiplication is 1-3 ╳ 1 56544.

Example 3

Decomposition 5x 2+6xy -8y 2. Analysis: This polynomial can be regarded as a quadratic trinomial about x, and-8y 2 is a constant term. When we decompose the coefficients of quadratic terms and constant terms, we only need to decompose 5 and -8, and then use the crosshairs to decompose them. After observation, we can choose a suitable group, namely120.

Example 4

Factorization (x-y)(2x-2y-3)-2. Analysis: This polynomial is the product of two factors and the difference of another factor. Only multiply the polynomials first, and then factorize the deformed polynomials. Q: What are the characteristics of the factors of the above product? What is the simplest method of polynomial multiplication? Answer: If we put forward the common factor 2 for the first two terms in the second factor, it will become 2(x-y), which is twice that of the first factor, and then multiply (x-y) as a whole, we can transform the original polynomial into a quadratic trinomial about (x-y). Cross multiplication can be used to decompose factors. The solution (x-y) (2x-2y-3)-2 = (x-y) [2 (x-y)-3]-2 = 2 (x-y) 2-3 (x-y)-21-2. =(x-y-2)(2x-2y+ 1)。 It is pointed out that decomposing (x-y) into a whole is another application of the "whole" thinking method in mathematics.

Example 5

X 2+2x- 15 analysis: constant term (-15)

Common methods for editing this paragraph

Firstly, the quadratic term is decomposed into (1 X quadratic term coefficient), and the constant term is decomposed into (1 X constant term). Then the constant term of quadratic term coefficient is written in the following format: if the value is equal to the coefficient of linear term after cross multiplication, it is true; If they are not equal, the test shall be conducted according to the following methods. (The general question is very simple, and the correct answer can be worked out at most 3 times. The format that needs many experiments is: (Note: abcd at this time does not refer to the coefficient in (AX 2+BX+C). And abcd is preferably an integer) a b ╳ c d First time a= 1 b= 1 c= quadratic term coefficient ÷a d= constant term ÷b Second time a= 1 b=2 c= quadratic term coefficient ÷ a d. Number ÷a d= constant term ÷b Fifth a=2 b=3 c= quadratic coefficient ÷a d= constant term ÷b Sixth a=3 b=2 c= quadratic coefficient ÷a d= constant term ÷b Seventh A = 3 b = 3 c.

Edit this cross product (solve the problem of the ratio between the two)

principle

Individuals in a set have only two different values, some of which are A and the others are B. The average value is C. Find the ratio of individuals with value A to individuals with value B.. Assuming that the total quantity is S, the number of A is M and B is s-m, then: [A * M+B * (S-M)]/S = Ca/S * M/S+B/S * (S-M)/S = cm/S = (C-B)/(A-) Divide the two as mentioned above = each b gets several values given by a ... that is, the proportion is more clear in the form of cross multiplication.

Pay attention when using cross multiplication.

The first point: it is used to solve the problem of the ratio of the two. The second point: the obtained proportional relationship is the proportional relationship of cardinal number. The third point: put the total average in the middle, on the diagonal, reduce the large number and put the result on the diagonal.

example

There were 7,650 college graduates in 2006, an increase of 2% over the previous year. Among them, the number of undergraduate graduates decreased by 2% compared with the previous year, and the number of graduate graduates increased by 10%. So, how many undergraduates have graduated from this university this year (2006)? Cross multiplication: 7500,7650 graduates last year ÷ (1+2%) = 7500. Undergraduates:-2% ............................................... 2% Postgraduates: 10%...-4% Undergraduates: Postgraduates = 8%: (-4%) =-2: 1. Last year's undergraduate students: 7500×2/3=5000 This year's undergraduate students: 5000×0.98=4900 A: There are 4900 undergraduates graduated from this university this year. The problem of chickens and rabbits in the same cage today, chickens and rabbits are in the same cage, with 35 heads above and 94 feet below. What are the geometric shapes of pheasants and rabbits? Solution of cross multiplication: assume that all chicken feet have 70 feet and all rabbit feet have 120 feet. Chickens: 70

Edit paragraph 3. Solving quadratic equation with one variable by cross multiplication.

Example 1 Factorization 2x 2-7x+3. Analysis: First, the quadratic term coefficient is decomposed and written in the upper left corner and the lower left corner of the crosshair, then the constant term is decomposed and written in the upper right corner and the lower right corner of the crosshair, and then the algebraic sum is obtained by cross multiplication to make it equal to the linear term coefficient. Decomposition of quadratic coefficient (positive factor only): 2 = 650. Decomposition constant term: 3 =1× 3 = 3 ×1= (-3) × (-1) × (-3) The following four cases are represented by crossing lines:11╳ 20. =-51-3 ╳ 2-11)+2× (-3) = -7 After observation, the fourth case is correct, because after cross multiplication, the sum of the two terms is exactly equal to the coefficient of the first term. For the quadratic trinomial AX 2+BX+C (A ≠ 0), if the quadratic coefficient A can be decomposed into the product of two factors, that is, a=a 1a2, the constant term C can be decomposed into the product of two factors, that is, c=c 1c2 and a 1 a2, C the arrangement is as follows: a1c1╳ a2c2a1c2c6 If it is exactly equal to the first-order coefficient b of the quadratic trinomial AX2+BX+C, then the quadratic trinomial can be decomposed into two factors A1x+c/kloc-. Example 2 Decomposition of 6x 2-7x -5. There are eight different permutation methods, among which 21╳ 3-52× (-5)+3×1=-7 is correct, so the original polynomial can be factorized by cross multiplication, and the solution is 6x2-7x-5 = (2x+ 1). When a quadratic trinomial factor whose quadratic coefficient is not 1 is decomposed by cross multiplication, it is often necessary to observe it many times to determine whether the factor can be decomposed by cross multiplication. For the quadratic trinomial with quadratic term coefficient of 1, the factor can also be decomposed by cross multiplication. At this time, we only need to consider how to decompose the constant term. For example, X 2+2x- 15 can be used as the decomposition factor. The cross multiplication is1-3 ╳151× 5+1× (-3) = 2, so x 2+2x- 15 = (x-3) (x+5 When we decompose the coefficients of quadratic term and constant term, we only need to decompose 5 and -8. After cross-hair decomposition, we can choose a suitable group through observation, that is,12 ╳ 5-41× (-4)+5× 2 = 6 to solve 5x2+6xy-8y2 = Y. Only multiply the polynomials first, and then factorize the deformed polynomials. Q: What are the characteristics of the factor of the product of two factors? What is the simplest method of polynomial multiplication? Answer: If we put forward the common factor 2 for the first two terms in the second factor, it will become 2(x-y), which is twice that of the first factor, and then multiply (x-y) as a whole, we can transform the original polynomial into a quadratic trinomial about (x-y). Cross multiplication can be used to decompose factors. The solution (x-y) (2x-2y-3)-2 = (x-y) [2 (x-y)-3]-2 = 2 (x-y) 2-3 (x-y)-21-2. =(x-y-2)(2x-2y+ 1)。 It is pointed out that factorization takes (x-y) as a whole, which uses the "whole" thinking method in mathematics. Example 5 x 2+2x- 15 analysis: the constant term (-65438) 0 can be decomposed into the product of two numbers with different signs, and can be decomposed into (-1)( 15) or (1) (-/kloc). =(x-3)(x+5) Summary: ① Factorization of formula x 2+(p+q) x+PQ The characteristics of this kind of quadratic trinomial are: the coefficient of quadratic term is1; Constant term is the product of two numbers; The coefficient of a linear term is the sum of two factors of a constant term. So we can directly decompose some quadratic trinomials with the coefficient of1:x2+(p+q) x+pq = (x+p) (x+q) 2kx2+MX+n into kx2+MX+n = (ax+b) (CX+d) ab ╳. =-8 Simplified arrangement X 2-3x- 10 = 0 (the left side of the equation is a quadratic trinomial and the right side is zero) (x-5)(x+2)=0 (the factorization factor on the left side of the equation) ∴x-5=0 or x+2=0 (converted into two linear equations). It should be remembered that there are two solutions to the quadratic equation of one variable. (3) Solution: 6x2+5x-50 = 0 (2x-5) (3x+10) = 0 (pay special attention to the symbols when factorizing factors by cross multiplication) ∴2x-5=0 or 3x+10 = 0 ∴. . (4) solution: x2-2 (+) x+4 = 0 (∵ 4 can be decomposed into 2.2, ∴ this problem can be factorized) (x-2) (x-2) = 0 ∴ x 1 = 2, x 2 = example x 2.