document.write( "Question 278949: please help me solve this 5n(squared+2n+6=0 \n" ); document.write( "

Algebra.Com's Answer #202887 by richwmiller(17219) $\"\"$ $\"About$

You can put this solution on YOUR website!
5n^2+2n+6=0
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Solved by pluggable solver: Factoring using the AC method (Factor by Grouping)

Looking at the expression $\"5n%5E2%2B2n%2B6\"$ , we can see that the first coefficient is $\"5\"$ , the second coefficient is $\"2\"$ , and the last term is $\"6\"$ .

Now multiply the first coefficient $\"5\"$ by the last term $\"6\"$ to get $\"%285%29%286%29=30\"$ .

Now the question is: what two whole numbers multiply to $\"30\"$ (the previous product) and add to the second coefficient $\"2\"$ ?

To find these two numbers, we need to list all of the factors of $\"30\"$ (the previous product).

Factors of $\"30\"$ :

1,2,3,5,6,10,15,30

-1,-2,-3,-5,-6,-10,-15,-30

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

These factors pair up and multiply to $\"30\"$ .

1*30 = 30
2*15 = 30
3*10 = 30
5*6 = 30
(-1)*(-30) = 30
(-2)*(-15) = 30
(-3)*(-10) = 30
(-5)*(-6) = 30

Now let's add up each pair of factors to see if one pair adds to the middle coefficient $\"2\"$ :

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First Number	Second Number	Sum
1	30	1+30=31
2	15	2+15=17
3	10	3+10=13
5	6	5+6=11
-1	-30	-1+(-30)=-31
-2	-15	-2+(-15)=-17
-3	-10	-3+(-10)=-13
-5	-6	-5+(-6)=-11

From the table, we can see that there are no pairs of numbers which add to $\"2\"$ . So $\"5n%5E2%2B2n%2B6\"$ cannot be factored.

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Answer:

So $\"5%2An%5E2%2B2%2An%2B6\"$ doesn't factor at all (over the rational numbers).

So $\"5%2An%5E2%2B2%2An%2B6\"$ is prime.

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Solved by pluggable solver: SOLVE quadratic equation with variable

Quadratic equation $\"an%5E2%2Bbn%2Bc=0\"$ (in our case $\"5n%5E2%2B2n%2B6+=+0\"$ ) has the following solutons:
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\n" ); document.write( " $\"n%5B12%5D+=+%28b%2B-sqrt%28+b%5E2-4ac+%29%29%2F2%5Ca\"$
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\n" ); document.write( " For these solutions to exist, the discriminant $\"b%5E2-4ac\"$ should not be a negative number.
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\n" ); document.write( " First, we need to compute the discriminant $\"b%5E2-4ac\"$ : $\"b%5E2-4ac=%282%29%5E2-4%2A5%2A6=-116\"$ .
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\n" ); document.write( " The discriminant -116 is less than zero. That means that there are no solutions among real numbers.

\n" ); document.write( " If you are a student of advanced school algebra and are aware about imaginary numbers, read on.

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\n" ); document.write( " In the field of imaginary numbers, the square root of -116 is + or - $\"sqrt%28+116%29+=+10.770329614269\"$ .
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\n" ); document.write( " The solution is

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\n" ); document.write( " Here's your graph:
\n" ); document.write( " $\"graph%28+500%2C+500%2C+-10%2C+10%2C+-20%2C+20%2C+5%2Ax%5E2%2B2%2Ax%2B6+%29\"$

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