document.write( "Question 260018: \"The square of a positive number is six more than five times the positive number. Find the number.\"
\n" ); document.write( "I'm confused because we did not do this type of problem as an example in class.
\n" ); document.write( "We are required to write a let statement. So i have:
\n" ); document.write( "Let n=the positive number
\n" ); document.write( "and then from my let statement and the word problem given, i took a guess and did this:
\n" ); document.write( "n(squared) = 5n + 6
\n" ); document.write( "Is this correct? And if so... how do i go on solving it? \n" ); document.write( "

Algebra.Com's Answer #191457 by richwmiller(17219) $\"\"$ $\"About$

You can put this solution on YOUR website!
n^2=5n+6
\n" ); document.write( "Looks good
\n" ); document.write( "n^2-5n-6=0
\n" ); document.write( "find the factors
\n" ); document.write( "set the factors equal to 0
\n" ); document.write( "and solve the two factors for n
\n" ); document.write( "one will be positive and one will be negative you wan the positive x (The factor will have a minus in it)
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Solved by pluggable solver: Factoring using the AC method (Factor by Grouping)

Looking at the expression $\"n%5E2-5n-6\"$ , we can see that the first coefficient is $\"1\"$ , the second coefficient is $\"-5\"$ , and the last term is $\"-6\"$ .

Now multiply the first coefficient $\"1\"$ by the last term $\"-6\"$ to get $\"%281%29%28-6%29=-6\"$ .

Now the question is: what two whole numbers multiply to $\"-6\"$ (the previous product) and add to the second coefficient $\"-5\"$ ?

To find these two numbers, we need to list all of the factors of $\"-6\"$ (the previous product).

Factors of $\"-6\"$ :

1,2,3,6

-1,-2,-3,-6

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

These factors pair up and multiply to $\"-6\"$ .

1*(-6) = -6
2*(-3) = -6
(-1)*(6) = -6
(-2)*(3) = -6

Now let's add up each pair of factors to see if one pair adds to the middle coefficient $\"-5\"$ :

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First Number	Second Number	Sum
1	-6	1+(-6)=-5
2	-3	2+(-3)=-1
-1	6	-1+6=5
-2	3	-2+3=1

From the table, we can see that the two numbers $\"1\"$ and $\"-6\"$ add to $\"-5\"$ (the middle coefficient).

So the two numbers $\"1\"$ and $\"-6\"$ both multiply to $\"-6\"$ and add to $\"-5\"$

Now replace the middle term $\"-5n\"$ with $\"n-6n\"$ . Remember, $\"1\"$ and $\"-6\"$ add to $\"-5\"$ . So this shows us that $\"n-6n=-5n\"$ .

$\"n%5E2%2Bhighlight%28n-6n%29-6\"$ Replace the second term $\"-5n\"$ with $\"n-6n\"$ .

$\"%28n%5E2%2Bn%29%2B%28-6n-6%29\"$ Group the terms into two pairs.

$\"n%28n%2B1%29%2B%28-6n-6%29\"$ Factor out the GCF $\"n\"$ from the first group.

$\"n%28n%2B1%29-6%28n%2B1%29\"$ Factor out $\"6\"$ 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.

$\"%28n-6%29%28n%2B1%29\"$ Combine like terms. Or factor out the common term $\"n%2B1\"$

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Answer:

So $\"n%5E2-5%2An-6\"$ factors to $\"%28n-6%29%28n%2B1%29\"$ .

In other words, $\"n%5E2-5%2An-6=%28n-6%29%28n%2B1%29\"$ .

Note: you can check the answer by expanding $\"%28n-6%29%28n%2B1%29\"$ to get $\"n%5E2-5%2An-6\"$ or by graphing the original expression and the answer (the two graphs should be identical).

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