document.write( "Question 615919: factor.......\r
\n" ); document.write( "\n" ); document.write( "3A(squared)-22a+7\r
\n" ); document.write( "\n" ); document.write( "next problem...\r
\n" ); document.write( "\n" ); document.write( "simplify \r
\n" ); document.write( "\n" ); document.write( "4m+20 over 4m(squared) - 100 \n" ); document.write( "

Algebra.Com's Answer #387427 by jim_thompson5910(35256) $\"\"$ $\"About$

You can put this solution on YOUR website!
I'll do the first one to get you started\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "Looking at the expression $\"3a%5E2-22a%2B7\"$ , we can see that the first coefficient is $\"3\"$ , the second coefficient is $\"-22\"$ , and the last term is $\"7\"$ .\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "Now multiply the first coefficient $\"3\"$ by the last term $\"7\"$ to get $\"%283%29%287%29=21\"$ .\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "Now the question is: what two whole numbers multiply to $\"21\"$ (the previous product) and add to the second coefficient $\"-22\"$ ?\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 $\"21\"$ (the previous product).\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "Factors of $\"21\"$ :\r
\n" ); document.write( "\n" ); document.write( "1,3,7,21\r
\n" ); document.write( "\n" ); document.write( "-1,-3,-7,-21\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 $\"21\"$ .\r
\n" ); document.write( "\n" ); document.write( "1*21 = 21
\n" ); document.write( "3*7 = 21
\n" ); document.write( "(-1)*(-21) = 21
\n" ); document.write( "(-3)*(-7) = 21\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 $\"-22\"$ :\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	21	1+21=22
3	7	3+7=10
-1	-21	-1+(-21)=-22
-3	-7	-3+(-7)=-10

\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 $\"-21\"$ add to $\"-22\"$ (the middle coefficient).\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "So the two numbers $\"-1\"$ and $\"-21\"$ both multiply to $\"21\"$ and add to $\"-22\"$ \r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "Now replace the middle term $\"-22a\"$ with $\"-a-21a\"$ . Remember, $\"-1\"$ and $\"-21\"$ add to $\"-22\"$ . So this shows us that $\"-a-21a=-22a\"$ .\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( " $\"3a%5E2%2Bhighlight%28-a-21a%29%2B7\"$ Replace the second term $\"-22a\"$ with $\"-a-21a\"$ .\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( " $\"%283a%5E2-a%29%2B%28-21a%2B7%29\"$ Group the terms into two pairs.\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( " $\"a%283a-1%29%2B%28-21a%2B7%29\"$ Factor out the GCF $\"a\"$ from the first group.\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( " $\"a%283a-1%29-7%283a-1%29\"$ 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( " $\"%28a-7%29%283a-1%29\"$ Combine like terms. Or factor out the common term $\"3a-1\"$ \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 $\"3a%5E2-22a%2B7\"$ factors to $\"%28a-7%29%283a-1%29\"$ .\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "In other words, $\"3a%5E2-22a%2B7=%28a-7%29%283a-1%29\"$ .\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "Note: you can check the answer by expanding $\"%28a-7%29%283a-1%29\"$ to get $\"3a%5E2-22a%2B7\"$ or by graphing the original expression and the answer (the two graphs should be identical).
\n" ); document.write( " \n" ); document.write( "

\n" );