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Question 903794: When the sum of 348 and three times a positive number is subtracted from the square of the number, the result is 112. Find the number.
Answer by richwmiller(17219)   (Show Source):
You can put this solution on YOUR website! n^2-(348+3n)=112
n^2-348-3n=112
n^2-348-3n-112=0
n^2-3n-460=0
| Solved by pluggable solver: Factoring using the AC method (Factor by Grouping) |
Looking at the expression  , we can see that the first coefficient is  , the second coefficient is  , and the last term is  .
Now multiply the first coefficient by the last term to get  .
Now the question is: what two whole numbers multiply to (the previous product) and add to the second coefficient ?
To find these two numbers, we need to list all of the factors of (the previous product).
Factors of :
1,2,4,5,10,20,23,46,92,115,230,460
-1,-2,-4,-5,-10,-20,-23,-46,-92,-115,-230,-460
Note: list the negative of each factor. This will allow us to find all possible combinations.
These factors pair up and multiply to .
1*(-460) = -460
2*(-230) = -460
4*(-115) = -460
5*(-92) = -460
10*(-46) = -460
20*(-23) = -460
(-1)*(460) = -460
(-2)*(230) = -460
(-4)*(115) = -460
(-5)*(92) = -460
(-10)*(46) = -460
(-20)*(23) = -460
Now let's add up each pair of factors to see if one pair adds to the middle coefficient :
| First Number | Second Number | Sum | | 1 | -460 | 1+(-460)=-459 | | 2 | -230 | 2+(-230)=-228 | | 4 | -115 | 4+(-115)=-111 | | 5 | -92 | 5+(-92)=-87 | | 10 | -46 | 10+(-46)=-36 | | 20 | -23 | 20+(-23)=-3 | | -1 | 460 | -1+460=459 | | -2 | 230 | -2+230=228 | | -4 | 115 | -4+115=111 | | -5 | 92 | -5+92=87 | | -10 | 46 | -10+46=36 | | -20 | 23 | -20+23=3 |
From the table, we can see that the two numbers  and  add to (the middle coefficient).
So the two numbers and both multiply to and add to
Now replace the middle term  with  . Remember, and add to . So this shows us that  .
 Replace the second term with .
 Group the terms into two pairs.
 Factor out the GCF  from the first group.
 Factor out  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.
 Combine like terms. Or factor out the common term 
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Answer:
So  factors to .
In other words,  .
Note: you can check the answer by expanding to get or by graphing the original expression and the answer (the two graphs should be identical).
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| Solved by pluggable solver: COMPLETING THE SQUARE solver for quadratics |
Read this lesson on completing the square by prince_abubu, if you do not know how to complete the square.
Let's convert  to standard form by dividing both sides by 1:
We have: .
What we want to do now is to change this equation to a complete square  . How can we find out values of somenumber and othernumber that would make it work?
Look at  :  . Since the coefficient in our equation  that goes in front of n is -3, we know that -3=2*somenumber, or  . So, we know that our equation can be rewritten as  , and we do not yet know the other number.
We are almost there. Finding the other number is simply a matter of not making too many mistakes. We need to find 'other number' such that is equivalent to our original equation  .
The highlighted red part must be equal to -460 (highlighted green part).
 , or  .
So, the equation converts to  , or  .
Our equation converted to a square  , equated to a number (462.25).
Since the right part 462.25 is greater than zero, there are two solutions:
, or
Answer: n=23, -20.
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