SOLUTION: Please help me solve this problem. factor completly:7x^2+19x-36

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Question 110334: Please help me solve this problem. factor completly:7x^2+19x-36
Found 2 solutions by scott8148, jim_thompson5910:
Answer by scott8148(6628) About Me  (Show Source):
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 7x%5E2%2B19x-36, we can see that the first coefficient is 7, the second coefficient is 19, and the last term is -36.



Now multiply the first coefficient 7 by the last term -36 to get %287%29%28-36%29=-252.



Now the question is: what two whole numbers multiply to -252 (the previous product) and add to the second coefficient 19?



To find these two numbers, we need to list all of the factors of -252 (the previous product).



Factors of -252:

1,2,3,4,6,7,9,12,14,18,21,28,36,42,63,84,126,252

-1,-2,-3,-4,-6,-7,-9,-12,-14,-18,-21,-28,-36,-42,-63,-84,-126,-252



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



These factors pair up and multiply to -252.

1*(-252) = -252
2*(-126) = -252
3*(-84) = -252
4*(-63) = -252
6*(-42) = -252
7*(-36) = -252
9*(-28) = -252
12*(-21) = -252
14*(-18) = -252
(-1)*(252) = -252
(-2)*(126) = -252
(-3)*(84) = -252
(-4)*(63) = -252
(-6)*(42) = -252
(-7)*(36) = -252
(-9)*(28) = -252
(-12)*(21) = -252
(-14)*(18) = -252


Now let's add up each pair of factors to see if one pair adds to the middle coefficient 19:



First NumberSecond NumberSum
1-2521+(-252)=-251
2-1262+(-126)=-124
3-843+(-84)=-81
4-634+(-63)=-59
6-426+(-42)=-36
7-367+(-36)=-29
9-289+(-28)=-19
12-2112+(-21)=-9
14-1814+(-18)=-4
-1252-1+252=251
-2126-2+126=124
-384-3+84=81
-463-4+63=59
-642-6+42=36
-736-7+36=29
-928-9+28=19
-1221-12+21=9
-1418-14+18=4




From the table, we can see that the two numbers -9 and 28 add to 19 (the middle coefficient).



So the two numbers -9 and 28 both multiply to -252 and add to 19



Now replace the middle term 19x with -9x%2B28x. Remember, -9 and 28 add to 19. So this shows us that -9x%2B28x=19x.



7x%5E2%2Bhighlight%28-9x%2B28x%29-36 Replace the second term 19x with -9x%2B28x.



%287x%5E2-9x%29%2B%2828x-36%29 Group the terms into two pairs.



x%287x-9%29%2B%2828x-36%29 Factor out the GCF x from the first group.



x%287x-9%29%2B4%287x-9%29 Factor out 4 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.



%28x%2B4%29%287x-9%29 Combine like terms. Or factor out the common term 7x-9



===============================================================



Answer:



So 7%2Ax%5E2%2B19%2Ax-36 factors to %28x%2B4%29%287x-9%29.



In other words, 7%2Ax%5E2%2B19%2Ax-36=%28x%2B4%29%287x-9%29.



Note: you can check the answer by expanding %28x%2B4%29%287x-9%29 to get 7%2Ax%5E2%2B19%2Ax-36 or by graphing the original expression and the answer (the two graphs should be identical).