document.write( "Question 110996: Please help me with this problem. Factor completely: 6x^2-25x+24 \n" ); document.write( "

Algebra.Com's Answer #80973 by jim_thompson5910(35256) $\"\"$ $\"About$

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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 $\"-25\"$ :

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First Number	Second Number	Sum
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
-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).

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