Cross multiplication is a common method to decompose quadratic trinomial. Firstly, it decomposes the quadratic coefficient A and the constant term C of the quadratic trinomial into the product of two factors (usually there are several different methods).
Then cross-multiply and add diagonally. If there is, then there is. Otherwise, try again where the exchange is needed. If it still doesn't work, change to another group and try in the same way until you find the right one.
When we do factorization, we can refer to the following formula:
First extract the common factor, then consider the formula;
Try cross multiplication, and the grouping should be appropriate;
Four methods are tried repeatedly, and the last one must be multiplication.
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 Handle 5x? +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 5x? +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 6x? -5x-25=0
Analysis: put 6x? If -5x-25 is regarded as a quadratic trinomial about X, then 6 can be divided into 1×6, 2×3 and -25 can be divided into-1×25, -5×5 and -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 14x? -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 9 years) (7 to 2 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.