SOLUTION: Please help me with this problem. Factor completely: 6x^2-25x+24

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Question 110996: Please help me with this problem. Factor completely: 6x^2-25x+24
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
Solved by pluggable solver: Factoring using the AC method (Factor by Grouping)


Looking at the expression 6x%5E2-25x%2B24, we can see that the first coefficient is 6, the second coefficient is -25, and the last term is 24.



Now multiply the first coefficient 6 by the last term 24 to get %286%29%2824%29=144.



Now the question is: what two whole numbers multiply to 144 (the previous product) and add to the second coefficient -25?



To find these two numbers, we need to list all of the factors of 144 (the previous product).



Factors of 144:

1,2,3,4,6,8,9,12,16,18,24,36,48,72,144

-1,-2,-3,-4,-6,-8,-9,-12,-16,-18,-24,-36,-48,-72,-144



Note: list the negative of each factor. This will allow us to find all possible combinations.



These factors pair up and multiply to 144.

1*144 = 144
2*72 = 144
3*48 = 144
4*36 = 144
6*24 = 144
8*18 = 144
9*16 = 144
12*12 = 144
(-1)*(-144) = 144
(-2)*(-72) = 144
(-3)*(-48) = 144
(-4)*(-36) = 144
(-6)*(-24) = 144
(-8)*(-18) = 144
(-9)*(-16) = 144
(-12)*(-12) = 144


Now let's add up each pair of factors to see if one pair adds to the middle coefficient -25:



First NumberSecond NumberSum
11441+144=145
2722+72=74
3483+48=51
4364+36=40
6246+24=30
8188+18=26
9169+16=25
121212+12=24
-1-144-1+(-144)=-145
-2-72-2+(-72)=-74
-3-48-3+(-48)=-51
-4-36-4+(-36)=-40
-6-24-6+(-24)=-30
-8-18-8+(-18)=-26
-9-16-9+(-16)=-25
-12-12-12+(-12)=-24




From the table, we can see that the two numbers -9 and -16 add to -25 (the middle coefficient).



So the two numbers -9 and -16 both multiply to 144 and add to -25



Now replace the middle term -25x with -9x-16x. Remember, -9 and -16 add to -25. So this shows us that -9x-16x=-25x.



6x%5E2%2Bhighlight%28-9x-16x%29%2B24 Replace the second term -25x with -9x-16x.



%286x%5E2-9x%29%2B%28-16x%2B24%29 Group the terms into two pairs.



3x%282x-3%29%2B%28-16x%2B24%29 Factor out the GCF 3x from the first group.



3x%282x-3%29-8%282x-3%29 Factor out 8 from the second group. The goal of this step is to make the terms in the second parenthesis equal to the terms in the first parenthesis.



%283x-8%29%282x-3%29 Combine like terms. Or factor out the common term 2x-3



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Answer:



So 6%2Ax%5E2-25%2Ax%2B24 factors to %283x-8%29%282x-3%29.



In other words, 6%2Ax%5E2-25%2Ax%2B24=%283x-8%29%282x-3%29.



Note: you can check the answer by expanding %283x-8%29%282x-3%29 to get 6%2Ax%5E2-25%2Ax%2B24 or by graphing the original expression and the answer (the two graphs should be identical).