document.write( "Question 903805: When the sum of 291 and four times a positive number is subtracted from the square of the number, the result is 105. Find the number. Thank you! \n" ); document.write( "
Algebra.Com's Answer #548370 by richwmiller(17219)\"\" \"About 
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n^2-(291+4n)=105
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Solved by pluggable solver: Factoring using the AC method (Factor by Grouping)


Looking at the expression \"n%5E2-4n-396\", we can see that the first coefficient is \"1\", the second coefficient is \"-4\", and the last term is \"-396\".



Now multiply the first coefficient \"1\" by the last term \"-396\" to get \"%281%29%28-396%29=-396\".



Now the question is: what two whole numbers multiply to \"-396\" (the previous product) and add to the second coefficient \"-4\"?



To find these two numbers, we need to list all of the factors of \"-396\" (the previous product).



Factors of \"-396\":

1,2,3,4,6,9,11,12,18,22,33,36,44,66,99,132,198,396

-1,-2,-3,-4,-6,-9,-11,-12,-18,-22,-33,-36,-44,-66,-99,-132,-198,-396



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



These factors pair up and multiply to \"-396\".

1*(-396) = -396
2*(-198) = -396
3*(-132) = -396
4*(-99) = -396
6*(-66) = -396
9*(-44) = -396
11*(-36) = -396
12*(-33) = -396
18*(-22) = -396
(-1)*(396) = -396
(-2)*(198) = -396
(-3)*(132) = -396
(-4)*(99) = -396
(-6)*(66) = -396
(-9)*(44) = -396
(-11)*(36) = -396
(-12)*(33) = -396
(-18)*(22) = -396


Now let's add up each pair of factors to see if one pair adds to the middle coefficient \"-4\":



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First NumberSecond NumberSum
1-3961+(-396)=-395
2-1982+(-198)=-196
3-1323+(-132)=-129
4-994+(-99)=-95
6-666+(-66)=-60
9-449+(-44)=-35
11-3611+(-36)=-25
12-3312+(-33)=-21
18-2218+(-22)=-4
-1396-1+396=395
-2198-2+198=196
-3132-3+132=129
-499-4+99=95
-666-6+66=60
-944-9+44=35
-1136-11+36=25
-1233-12+33=21
-1822-18+22=4




From the table, we can see that the two numbers \"18\" and \"-22\" add to \"-4\" (the middle coefficient).



So the two numbers \"18\" and \"-22\" both multiply to \"-396\" and add to \"-4\"



Now replace the middle term \"-4n\" with \"18n-22n\". Remember, \"18\" and \"-22\" add to \"-4\". So this shows us that \"18n-22n=-4n\".



\"n%5E2%2Bhighlight%2818n-22n%29-396\" Replace the second term \"-4n\" with \"18n-22n\".



\"%28n%5E2%2B18n%29%2B%28-22n-396%29\" Group the terms into two pairs.



\"n%28n%2B18%29%2B%28-22n-396%29\" Factor out the GCF \"n\" from the first group.



\"n%28n%2B18%29-22%28n%2B18%29\" Factor out \"22\" 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.



\"%28n-22%29%28n%2B18%29\" Combine like terms. Or factor out the common term \"n%2B18\"



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Answer:



So \"n%5E2-4%2An-396\" factors to \"%28n-22%29%28n%2B18%29\".



In other words, \"n%5E2-4%2An-396=%28n-22%29%28n%2B18%29\".



Note: you can check the answer by expanding \"%28n-22%29%28n%2B18%29\" to get \"n%5E2-4%2An-396\" or by graphing the original expression and the answer (the two graphs should be identical).

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