document.write( "Question 110334: Please help me solve this problem. factor completly:7x^2+19x-36 \n" ); document.write( "
Algebra.Com's Answer #80433 by jim_thompson5910(35256)\"\" \"About 
You can put this solution on YOUR website!
\n" ); document.write( "\n" ); document.write( " \n" ); document.write( "
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\":



\n" ); document.write( "
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).

\n" ); document.write( " \n" ); document.write( "
\n" );