document.write( "Question 176331: 4) y²+5y-24 \n" ); document.write( "

Algebra.Com's Answer #131413 by jim_thompson5910(35256) $\"\"$ $\"About$

You can put this solution on YOUR website!
I assume that you want to factor?\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "Looking at the expression $\"y%5E2%2B5y-24\"$ , we can see that the first coefficient is $\"1\"$ , the second coefficient is $\"5\"$ , and the last term is $\"-24\"$ .\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "Now multiply the first coefficient $\"1\"$ by the last term $\"-24\"$ to get $\"%281%29%28-24%29=-24\"$ .\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "Now the question is: what two whole numbers multiply to $\"-24\"$ (the previous product) and add to the second coefficient $\"5\"$ ?\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 $\"-24\"$ (the previous product).\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "Factors of $\"-24\"$ :\r
\n" ); document.write( "\n" ); document.write( "1,2,3,4,6,8,12,24\r
\n" ); document.write( "\n" ); document.write( "-1,-2,-3,-4,-6,-8,-12,-24\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 $\"-24\"$ .\r
\n" ); document.write( "\n" ); document.write( "1*(-24)
\n" ); document.write( "2*(-12)
\n" ); document.write( "3*(-8)
\n" ); document.write( "4*(-6)
\n" ); document.write( "(-1)*(24)
\n" ); document.write( "(-2)*(12)
\n" ); document.write( "(-3)*(8)
\n" ); document.write( "(-4)*(6)\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 $\"5\"$ :\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	-24	1+(-24)=-23
2	-12	2+(-12)=-10
3	-8	3+(-8)=-5
4	-6	4+(-6)=-2
-1	24	-1+24=23
-2	12	-2+12=10
-3	8	-3+8=5
-4	6	-4+6=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 $\"8\"$ add to $\"5\"$ (the middle coefficient).\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "So the two numbers $\"-3\"$ and $\"8\"$ both multiply to $\"-24\"$ and add to $\"5\"$ \r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "Now replace the middle term $\"5y\"$ with $\"-3y%2B8y\"$ . Remember, $\"-3\"$ and $\"8\"$ add to $\"5\"$ . So this shows us that $\"-3y%2B8y=5y\"$ .\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( " $\"y%5E2%2Bhighlight%28-3y%2B8y%29-24\"$ Replace the second term $\"5y\"$ with $\"-3y%2B8y\"$ .\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( " $\"%28y%5E2-3y%29%2B%288y-24%29\"$ Group the terms into two pairs.\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( " $\"y%28y-3%29%2B%288y-24%29\"$ Factor out the GCF $\"y\"$ from the first group.\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( " $\"y%28y-3%29%2B8%28y-3%29\"$ Factor out $\"8\"$ 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( " $\"%28y%2B8%29%28y-3%29\"$ Combine like terms. Or factor out the common term $\"y-3\"$ \r
\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 $\"y%5E2%2B5y-24\"$ factors to $\"%28y%2B8%29%28y-3%29\"$ .\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "Note: you can check the answer by FOILing $\"%28y%2B8%29%28y-3%29\"$ to get $\"y%5E2%2B5y-24\"$ or by graphing the original expression and the answer (the two graphs should be identical).
\n" ); document.write( " \n" ); document.write( "

\n" );