document.write( "Question 862530: Find two consectuive Integers such that 15 times the difference of their reciprocols gives 2 \n" ); document.write( "

Algebra.Com's Answer #519696 by richwmiller(17219) $\"\"$ $\"About$

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Solved by pluggable solver: Factoring using the AC method (Factor by Grouping)

Looking at the expression $\"2x%5E2%2B2x-15\"$ , we can see that the first coefficient is $\"2\"$ , the second coefficient is $\"2\"$ , and the last term is $\"-15\"$ .

Now multiply the first coefficient $\"2\"$ by the last term $\"-15\"$ to get $\"%282%29%28-15%29=-30\"$ .

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)=-29
2	-15	2+(-15)=-13
3	-10	3+(-10)=-7
5	-6	5+(-6)=-1
-1	30	-1+30=29
-2	15	-2+15=13
-3	10	-3+10=7
-5	6	-5+6=1

From the table, we can see that there are no pairs of numbers which add to $\"2\"$ . So $\"2x%5E2%2B2x-15\"$ cannot be factored.

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Answer:

So $\"2%2Ax%5E2%2B2%2Ax-15\"$ doesn't factor at all (over the rational numbers).

So $\"2%2Ax%5E2%2B2%2Ax-15\"$ is prime.

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Solved by pluggable solver: SOLVE quadratic equation with variable

Quadratic equation $\"ax%5E2%2Bbx%2Bc=0\"$ (in our case $\"2x%5E2%2B2x%2B-15+=+0\"$ ) has the following solutons:
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\n" ); document.write( " $\"x%5B12%5D+=+%28b%2B-sqrt%28+b%5E2-4ac+%29%29%2F2%5Ca\"$
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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 $\"b%5E2-4ac\"$ : $\"b%5E2-4ac=%282%29%5E2-4%2A2%2A-15=124\"$ .
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\n" ); document.write( " Discriminant d=124 is greater than zero. That means that there are two solutions: $\"+x%5B12%5D+=+%28-2%2B-sqrt%28+124+%29%29%2F2%5Ca\"$ .
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\n" ); document.write( " $\"x%5B1%5D+=+%28-%282%29%2Bsqrt%28+124+%29%29%2F2%5C2+=+2.28388218141501\"$
\n" ); document.write( " $\"x%5B2%5D+=+%28-%282%29-sqrt%28+124+%29%29%2F2%5C2+=+-3.28388218141501\"$
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\n" ); document.write( " Quadratic expression $\"2x%5E2%2B2x%2B-15\"$ can be factored:
\n" ); document.write( " $\"2x%5E2%2B2x%2B-15+=+2%28x-2.28388218141501%29%2A%28x--3.28388218141501%29\"$
\n" ); document.write( " Again, the answer is: 2.28388218141501, -3.28388218141501.\n" ); document.write( "Here's your graph:
\n" ); document.write( " $\"graph%28+500%2C+500%2C+-10%2C+10%2C+-20%2C+20%2C+2%2Ax%5E2%2B2%2Ax%2B-15+%29\"$

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