SOLUTION: 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!

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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!
Answer by richwmiller(17219) About Me  (Show Source):
You can put this solution on YOUR website!
n^2-(291+4n)=105
n^2-291-4n-105=0
n^2-4n-396=0
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:



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



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



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).