document.write( "Question 483520: 5x^2-34x-7 \n" ); document.write( "

Algebra.Com's Answer #330887 by jim_thompson5910(35256) $\"\"$ $\"About$

You can put this solution on YOUR website!
\r
\n" ); document.write( "\n" ); document.write( "Looking at the expression $\"5x%5E2-34x-7\"$ , we can see that the first coefficient is $\"5\"$ , the second coefficient is $\"-34\"$ , and the last term is $\"-7\"$ .\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "Now multiply the first coefficient $\"5\"$ by the last term $\"-7\"$ to get $\"%285%29%28-7%29=-35\"$ .\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "Now the question is: what two whole numbers multiply to $\"-35\"$ (the previous product) and add to the second coefficient $\"-34\"$ ?\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "To find these two numbers, we need to list all of the factors of $\"-35\"$ (the previous product).\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "Factors of $\"-35\"$ :\r
\n" ); document.write( "\n" ); document.write( "1,5,7,35\r
\n" ); document.write( "\n" ); document.write( "-1,-5,-7,-35\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "Note: list the negative of each factor. This will allow us to find all possible combinations.\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "These factors pair up and multiply to $\"-35\"$ .\r
\n" ); document.write( "\n" ); document.write( "1*(-35) = -35
\n" ); document.write( "5*(-7) = -35
\n" ); document.write( "(-1)*(35) = -35
\n" ); document.write( "(-5)*(7) = -35\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "Now let's add up each pair of factors to see if one pair adds to the middle coefficient $\"-34\"$ :\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "\n" ); document.write( "\n" ); document.write( "

First Number	Second Number	Sum
1	-35	1+(-35)=-34
5	-7	5+(-7)=-2
-1	35	-1+35=34
-5	7	-5+7=2

\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "From the table, we can see that the two numbers $\"1\"$ and $\"-35\"$ add to $\"-34\"$ (the middle coefficient).\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "So the two numbers $\"1\"$ and $\"-35\"$ both multiply to $\"-35\"$ and add to $\"-34\"$ \r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "Now replace the middle term $\"-34x\"$ with $\"x-35x\"$ . Remember, $\"1\"$ and $\"-35\"$ add to $\"-34\"$ . So this shows us that $\"x-35x=-34x\"$ .\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( " $\"5x%5E2%2Bhighlight%28x-35x%29-7\"$ Replace the second term $\"-34x\"$ with $\"x-35x\"$ .\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( " $\"%285x%5E2%2Bx%29%2B%28-35x-7%29\"$ Group the terms into two pairs.\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( " $\"x%285x%2B1%29%2B%28-35x-7%29\"$ Factor out the GCF $\"x\"$ from the first group.\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( " $\"x%285x%2B1%29-7%285x%2B1%29\"$ Factor out $\"7\"$ 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.\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( " $\"%28x-7%29%285x%2B1%29\"$ Combine like terms. Or factor out the common term $\"5x%2B1\"$ \r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "===============================================================\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "Answer:\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "So $\"5x%5E2-34x-7\"$ factors to $\"%28x-7%29%285x%2B1%29\"$ .\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "In other words, $\"5x%5E2-34x-7=%28x-7%29%285x%2B1%29\"$ .\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "Note: you can check the answer by expanding $\"%28x-7%29%285x%2B1%29\"$ to get $\"5x%5E2-34x-7\"$ or by graphing the original expression and the answer (the two graphs should be identical).
\n" ); document.write( " \n" ); document.write( "

\n" );