1, the method of cross multiplication: the left multiplication of the cross is equal to the quadratic term coefficient, the right multiplication is equal to the constant term, and the cross multiplication is equal to the linear term coefficient.
2. Use of cross multiplication: (1) Cross multiplication is used to decompose factors. (2) Solving the quadratic equation with one variable by cross multiplication.
Advantages of cross multiplication: it is faster to solve problems with cross multiplication, which can save time and is not easy to make mistakes.
4. Defects of cross multiplication: 1. Some problems are relatively simple to solve by cross multiplication, but not every problem is simply solved by cross multiplication. 2. Cross multiplication is only applicable to quadratic trinomial type problems. 3. Cross multiplication is more difficult to learn.
5. Examples of cross multiplication problem solving:
1), using cross multiplication to solve some simple and common problems.
Example 1 M? +4m- 12 factorization factor
Analysis: The constant term-12 in this question can be divided into-1× 12, -2×6, -3×4, -6×2,-12× 1 2.
Solution: Because 1 -2
1 ╳ 6
So m? +4m- 12=(m-2)(m+6)
Example 2 5? +6x-8 factorization factor
Analysis: In this question, 5 can be divided into 1× 5, and -8 can be divided into-1×8, -2×4, -4×2, -8× 1. When the coefficient of quadratic term is divided into 1×5 and the constant term is divided into -4×2, it is consistent with this question.
Solution: Because 1 2
5 ╳ -4
So 5? +6x-8=(x+2)(5x-4)
Example 3 solving equation x? -8x+ 15=0
Analysis: put x? -8x+ 15 is regarded as a quadratic trinomial about x, then 15 can be divided into 1× 15 and 3×5.
Solution: Because 1 -3
1 ╳ -5
So the original equation can be transformed into (x-3)(x-5)=0.
So x 1=3 x2=5.
Example 4. Solve equation 6? -5x-25=0
Analysis: put 6? If 5x-25 is regarded as a quadratic trinomial about X, then 6 can be divided into 1×6, 2×3, -25 can be divided into-1×25, -5×5, -25× 1.
Solution: Because 2 -5
3 ╳ 5
So the original equation can be changed to (2x-5)(3x+5)=0.
So x 1=5/2 x2=-5/3.
2) Use cross multiplication to solve some difficult problems.
Example 5 14? -67xy+ 18y？ Decomposition factor
Analysis: put 14x? -67xy+ 18y？ As a quadratic trinomial about X, 14 can be divided into 1× 14, 2×7, 18y? It can be divided into y. 18y, 2y.9y and 3y.6y
Solution: Because 2-9 years old
7 ╳ -2y
So 14x? -67xy+ 18y？ =(2 to 2 years) (7 to 9 years)
Example 6 10x? -27xy-28y？ -x+25y-3 factorization factor
Analysis: This question should organize this polynomial into a quadratic trinomial form.
Solution 1, 10x? -27xy-28y？ -x+25y-3
= 10x？ -(27y+ 1)x -(28y？ ; -25y+3)
4y -3
7y ╳ - 1
= 10x？ -(27y+ 1)x-(4y-3)(7y- 1)
=[2x-(7y- 1)][5x+(4y-3)]2-(7y– 1)
5 ╳ 4y - 3
=(2x -7y + 1)(5x +4y -3)
Note: this question, first put 28y? -25y+3 is decomposed into (4y-3)(7y-1), 10x? -(27y+1) x-(4y-3) (7y-1) is decomposed into [2x -(7y-1)][5x +(4y+3)].
Solution 2, 10x? -27xy-28y？ -x+25y-3
=(2x-7y)(5x+4y)-(x-25y)-3 ^ 2-7y
=[(2x-7y)+ 1][(5x-4y)-3]5╳4y
=(2x-7y+ 1)(5x-4y-3)2x-7y 1
5 x - 4y ╳ -3
Note: For this question, put 10x first. -27xy-28y？ It is decomposed into (2x -7y)(5x +4y) by cross multiplication, and then it is decomposed into [(2x -7y)+ 1] [(5x -4y)-3] by cross multiplication.
Example 7: Solve the equation about x: x? - 3ax + 2a？ -a B- b？ =0
Analysis: 2a? -a B- b？ Cross multiplication can be used for factorization.
Solution: x? - 3ax + 2a？ -a B- b？ =0
x？ - 3ax +(2a？ -a B- b？ )=0
x？ - 3ax +(2a+b)(a-b)=0 1 -b
2 ╳ +b
[x-(2a+b)][x-(a-b)]= 0 1-(2a+b)
1 ╳ -(a-b)
So x1= 2a+bx2 = a-b.
pay attention to
1. When factorizing some quadratic trinomials in the form of ax2+bx+c by cross multiplication, we should pay attention to the following problems:
(1) Correct cross multiplication must meet the following conditions:
a 1 c 1
In the formula? The two vertical numbers must satisfy the relationship a 1a2=a, c1C2 = c; In the above formula, oblique
a2 c2
Two numbers must satisfy the relation a1C2+a2c1= B.
(2) When writing two decomposed linear factors from four numbers in the cross multiplication graph, a 1 is the linear term coefficient in the first factor, and c 1 is the constant term in the top two numbers in the graph; In the next two numbers, a2 is the coefficient of the first term in the second factor, and c2 is the constant term.
(3) Generally, the quadratic coefficient A is regarded as a positive number (if it is negative, we need to put forward a negative sign and transform it into a positive number by invariant deformation), and we only need to decompose it into two positive factors.
2. Some quadratic trinomials in the form of x+px+q can also be factorized by cross multiplication.
3. Any polynomial that can be converted into quadratic trinomial ax+bx+c by method of substitution can also be factorized by cross multiplication, as shown in Example 4.