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

Algebra.Com's Answer #548328 by richwmiller(17219) $\"\"$ $\"About$

You can put this solution on YOUR website!
n^2-(348+3n)=112
\n" ); document.write( "n^2-348-3n=112
\n" ); document.write( "n^2-348-3n-112=0
\n" ); document.write( "n^2-3n-460=0
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Solved by pluggable solver: Factoring using the AC method (Factor by Grouping)

Looking at the expression $\"n%5E2-3n-460\"$ , we can see that the first coefficient is $\"1\"$ , the second coefficient is $\"-3\"$ , and the last term is $\"-460\"$ .

Now multiply the first coefficient $\"1\"$ by the last term $\"-460\"$ to get $\"%281%29%28-460%29=-460\"$ .

Now the question is: what two whole numbers multiply to $\"-460\"$ (the previous product) and add to the second coefficient $\"-3\"$ ?

To find these two numbers, we need to list all of the factors of $\"-460\"$ (the previous product).

Factors of $\"-460\"$ :

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 $\"-460\"$ .

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 $\"-3\"$ :

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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 $\"20\"$ and $\"-23\"$ add to $\"-3\"$ (the middle coefficient).

So the two numbers $\"20\"$ and $\"-23\"$ both multiply to $\"-460\"$ and add to $\"-3\"$

Now replace the middle term $\"-3n\"$ with $\"20n-23n\"$ . Remember, $\"20\"$ and $\"-23\"$ add to $\"-3\"$ . So this shows us that $\"20n-23n=-3n\"$ .

$\"n%5E2%2Bhighlight%2820n-23n%29-460\"$ Replace the second term $\"-3n\"$ with $\"20n-23n\"$ .

$\"%28n%5E2%2B20n%29%2B%28-23n-460%29\"$ Group the terms into two pairs.

$\"n%28n%2B20%29%2B%28-23n-460%29\"$ Factor out the GCF $\"n\"$ from the first group.

$\"n%28n%2B20%29-23%28n%2B20%29\"$ Factor out $\"23\"$ 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.

$\"%28n-23%29%28n%2B20%29\"$ Combine like terms. Or factor out the common term $\"n%2B20\"$

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Answer:

So $\"n%5E2-3%2An-460\"$ factors to $\"%28n-23%29%28n%2B20%29\"$ .

In other words, $\"n%5E2-3%2An-460=%28n-23%29%28n%2B20%29\"$ .

Note: you can check the answer by expanding $\"%28n-23%29%28n%2B20%29\"$ to get $\"n%5E2-3%2An-460\"$ 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 $\"1n%5E2%2B-3n%2B-460=0\"$ to standard form by dividing both sides by 1:
\n" ); document.write( "We have: $\"1n%5E2%2B-3n%2B-460=0\"$ . \n" ); document.write( "What we want to do now is to change this equation to a complete square $\"%28n%2Bsomenumber%29%5E2+%2B+othernumber\"$ . How can we find out values of somenumber and othernumber that would make it work?
\n" ); document.write( "Look at $\"%28n%2Bsomenumber%29%5E2\"$ : $\"%28n%2Bsomenumber%29%5E2+=+n%5E2%2B2%2Asomenumber%2Ax+%2B+somenumber%5E2\"$ . Since the coefficient in our equation $\"1n%5E2%2Bhighlight_red%28+-3%29+%2A+n%2B-460=0\"$ that goes in front of n is -3, we know that -3=2*somenumber, or $\"somenumber+=+-3%2F2\"$ . So, we know that our equation can be rewritten as $\"%28n%2B-3%2F2%29%5E2+%2B+othernumber\"$ , and we do not yet know the other number.
\n" ); document.write( "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 $\"%28n%2B-3%2F2%29%5E2+%2B+othernumber\"$ is equivalent to our original equation $\"1n%5E2%2B-3n%2Bhighlight_green%28+-460+%29=0\"$ .
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\n" ); document.write( " The highlighted red part must be equal to -460 (highlighted green part).
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\n" ); document.write( " $\"-3%5E2%2F4+%2B+othernumber+=+-460\"$ , or $\"othernumber+=+-460--3%5E2%2F4+=+-462.25\"$ .
\n" ); document.write( "So, the equation converts to $\"%28n%2B-3%2F2%29%5E2+%2B+-462.25+=+0\"$ , or $\"%28n%2B-3%2F2%29%5E2+=+462.25\"$ .
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\n" ); document.write( " Our equation converted to a square $\"%28n%2B-3%2F2%29%5E2\"$ , equated to a number (462.25).
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\n" ); document.write( " Since the right part 462.25 is greater than zero, there are two solutions:
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\n" ); document.write( " , or
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\n" ); document.write( " $\"system%28+%28n%2B-3%2F2%29+=+21.5%2C+%28n%2B-3%2F2%29+=+-21.5+%29\"$
\n" ); document.write( " $\"system%28+n%2B-3%2F2+=+21.5%2C+n%2B-3%2F2+=+-21.5+%29\"$
\n" ); document.write( " $\"system%28+n+=+21.5--3%2F2%2C+n+=+-21.5--3%2F2+%29\"$
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\n" ); document.write( " $\"system%28+n+=+23%2C+n+=+-20+%29\"$
\n" ); document.write( "Answer: n=23, -20.\n" ); document.write( "

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