document.write( "Question 644696: t^2-2t-15 \n" ); document.write( "

Algebra.Com's Answer #405213 by jim_thompson5910(35256) $\"\"$ $\"About$

You can put this solution on YOUR website!
I'm assuming you want to factor this.\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "Looking at the expression $\"t%5E2-2t-15\"$ , we can see that the first coefficient is $\"1\"$ , the second coefficient is $\"-2\"$ , and the last term is $\"-15\"$ .\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "Now multiply the first coefficient $\"1\"$ by the last term $\"-15\"$ to get $\"%281%29%28-15%29=-15\"$ .\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "Now the question is: what two whole numbers multiply to $\"-15\"$ (the previous product) and add to the second coefficient $\"-2\"$ ?\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 $\"-15\"$ (the previous product).\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "Factors of $\"-15\"$ :\r
\n" ); document.write( "\n" ); document.write( "1,3,5,15\r
\n" ); document.write( "\n" ); document.write( "-1,-3,-5,-15\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 $\"-15\"$ .\r
\n" ); document.write( "\n" ); document.write( "1*(-15) = -15
\n" ); document.write( "3*(-5) = -15
\n" ); document.write( "(-1)*(15) = -15
\n" ); document.write( "(-3)*(5) = -15\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 $\"-2\"$ :\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	-15	1+(-15)=-14
3	-5	3+(-5)=-2
-1	15	-1+15=14
-3	5	-3+5=2

\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "From the table, we can see that the two numbers $\"3\"$ and $\"-5\"$ add to $\"-2\"$ (the middle coefficient).\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "So the two numbers $\"3\"$ and $\"-5\"$ both multiply to $\"-15\"$ and add to $\"-2\"$ \r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "Now replace the middle term $\"-2t\"$ with $\"3t-5t\"$ . Remember, $\"3\"$ and $\"-5\"$ add to $\"-2\"$ . So this shows us that $\"3t-5t=-2t\"$ .\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( " $\"t%5E2%2Bhighlight%283t-5t%29-15\"$ Replace the second term $\"-2t\"$ with $\"3t-5t\"$ .\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( " $\"%28t%5E2%2B3t%29%2B%28-5t-15%29\"$ Group the terms into two pairs.\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( " $\"t%28t%2B3%29%2B%28-5t-15%29\"$ Factor out the GCF $\"t\"$ from the first group.\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( " $\"t%28t%2B3%29-5%28t%2B3%29\"$ Factor out $\"5\"$ 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( " $\"%28t-5%29%28t%2B3%29\"$ Combine like terms. Or factor out the common term $\"t%2B3\"$ \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 $\"t%5E2-2t-15\"$ factors to $\"%28t-5%29%28t%2B3%29\"$ .\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "In other words, $\"t%5E2-2t-15=%28t-5%29%28t%2B3%29\"$ .\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "Note: you can check the answer by expanding $\"%28t-5%29%28t%2B3%29\"$ to get $\"t%5E2-2t-15\"$ or by graphing the original expression and the answer (the two graphs should be identical).\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "--------------------------------------------------------------------------------------------------------------
\n" ); document.write( "If you need more help, email me at jim_thompson5910@hotmail.com\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "Also, please consider visiting my website: http://www.freewebs.com/jimthompson5910/home.html and making a donation. Thank you\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "Jim
\n" ); document.write( "-------------------------------------------------------------------------------------------------------------- \n" ); document.write( "

\n" );