document.write( "Question 729201: Factor by grouping. (If the expression is nonfactorable enter PRIME.)
\n" ); document.write( "50x2 – 50x – 12 \n" ); document.write( "

Algebra.Com's Answer #445764 by jim_thompson5910(35256) $\"\"$ $\"About$

You can put this solution on YOUR website!
\r
\n" ); document.write( "\n" ); document.write( " $\"50x%5E2-50x-12\"$ Start with the given expression.\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( " $\"2%2825x%5E2-25x-6%29\"$ Factor out the GCF $\"2\"$ .\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "Now let's try to factor the inner expression $\"25x%5E2-25x-6\"$ \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( "Looking at the expression $\"25x%5E2-25x-6\"$ , we can see that the first coefficient is $\"25\"$ , the second coefficient is $\"-25\"$ , and the last term is $\"-6\"$ .\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "Now multiply the first coefficient $\"25\"$ by the last term $\"-6\"$ to get $\"%2825%29%28-6%29=-150\"$ .\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "Now the question is: what two whole numbers multiply to $\"-150\"$ (the previous product) and add to the second coefficient $\"-25\"$ ?\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 $\"-150\"$ (the previous product).\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "Factors of $\"-150\"$ :\r
\n" ); document.write( "\n" ); document.write( "1,2,3,5,6,10,15,25,30,50,75,150\r
\n" ); document.write( "\n" ); document.write( "-1,-2,-3,-5,-6,-10,-15,-25,-30,-50,-75,-150\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 $\"-150\"$ .\r
\n" ); document.write( "\n" ); document.write( "1*(-150) = -150
\n" ); document.write( "2*(-75) = -150
\n" ); document.write( "3*(-50) = -150
\n" ); document.write( "5*(-30) = -150
\n" ); document.write( "6*(-25) = -150
\n" ); document.write( "10*(-15) = -150
\n" ); document.write( "(-1)*(150) = -150
\n" ); document.write( "(-2)*(75) = -150
\n" ); document.write( "(-3)*(50) = -150
\n" ); document.write( "(-5)*(30) = -150
\n" ); document.write( "(-6)*(25) = -150
\n" ); document.write( "(-10)*(15) = -150\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 $\"-25\"$ :\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	-150	1+(-150)=-149
2	-75	2+(-75)=-73
3	-50	3+(-50)=-47
5	-30	5+(-30)=-25
6	-25	6+(-25)=-19
10	-15	10+(-15)=-5
-1	150	-1+150=149
-2	75	-2+75=73
-3	50	-3+50=47
-5	30	-5+30=25
-6	25	-6+25=19
-10	15	-10+15=5

\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "From the table, we can see that the two numbers $\"5\"$ and $\"-30\"$ add to $\"-25\"$ (the middle coefficient).\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "So the two numbers $\"5\"$ and $\"-30\"$ both multiply to $\"-150\"$ and add to $\"-25\"$ \r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "Now replace the middle term $\"-25x\"$ with $\"5x-30x\"$ . Remember, $\"5\"$ and $\"-30\"$ add to $\"-25\"$ . So this shows us that $\"5x-30x=-25x\"$ .\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( " $\"25x%5E2%2Bhighlight%285x-30x%29-6\"$ Replace the second term $\"-25x\"$ with $\"5x-30x\"$ .\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( " $\"%2825x%5E2%2B5x%29%2B%28-30x-6%29\"$ Group the terms into two pairs.\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( " $\"5x%285x%2B1%29%2B%28-30x-6%29\"$ Factor out the GCF $\"5x\"$ from the first group.\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( " $\"5x%285x%2B1%29-6%285x%2B1%29\"$ Factor out $\"6\"$ 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( " $\"%285x-6%29%285x%2B1%29\"$ 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( "So $\"2%2825x%5E2-25x-6%29\"$ then factors further to $\"2%285x-6%29%285x%2B1%29\"$ \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 $\"50x%5E2-50x-12\"$ completely factors to $\"2%285x-6%29%285x%2B1%29\"$ .\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "In other words, $\"50x%5E2-50x-12=2%285x-6%29%285x%2B1%29\"$ .\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "Note: you can check the answer by expanding $\"2%285x-6%29%285x%2B1%29\"$ to get $\"50x%5E2-50x-12\"$ or by graphing the original expression and the answer (the two graphs should be identical).
\n" ); document.write( " \n" ); document.write( "

\n" );