document.write( "Question 186599: w^2-12w+20 \n" ); document.write( "

Algebra.Com's Answer #139861 by jim_thompson5910(35256) $\"\"$ $\"About$

You can put this solution on YOUR website!
\r
\n" ); document.write( "\n" ); document.write( "Looking at the expression $\"w%5E2-12w%2B20\"$ , we can see that the first coefficient is $\"1\"$ , the second coefficient is $\"-12\"$ , and the last term is $\"20\"$ .\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "Now multiply the first coefficient $\"1\"$ by the last term $\"20\"$ to get $\"%281%29%2820%29=20\"$ .\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "Now the question is: what two whole numbers multiply to $\"20\"$ (the previous product) and add to the second coefficient $\"-12\"$ ?\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 $\"20\"$ (the previous product).\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "Factors of $\"20\"$ :\r
\n" ); document.write( "\n" ); document.write( "1,2,4,5,10,20\r
\n" ); document.write( "\n" ); document.write( "-1,-2,-4,-5,-10,-20\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 $\"20\"$ .\r
\n" ); document.write( "\n" ); document.write( "1*20
\n" ); document.write( "2*10
\n" ); document.write( "4*5
\n" ); document.write( "(-1)*(-20)
\n" ); document.write( "(-2)*(-10)
\n" ); document.write( "(-4)*(-5)\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 $\"-12\"$ :\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	20	1+20=21
2	10	2+10=12
4	5	4+5=9
-1	-20	-1+(-20)=-21
-2	-10	-2+(-10)=-12
-4	-5	-4+(-5)=-9

\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "From the table, we can see that the two numbers $\"-2\"$ and $\"-10\"$ add to $\"-12\"$ (the middle coefficient).\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "So the two numbers $\"-2\"$ and $\"-10\"$ both multiply to $\"20\"$ and add to $\"-12\"$ \r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "Now replace the middle term $\"-12w\"$ with $\"-2w-10w\"$ . Remember, $\"-2\"$ and $\"-10\"$ add to $\"-12\"$ . So this shows us that $\"-2w-10w=-12w\"$ .\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( " $\"w%5E2%2Bhighlight%28-2w-10w%29%2B20\"$ Replace the second term $\"-12w\"$ with $\"-2w-10w\"$ .\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( " $\"%28w%5E2-2w%29%2B%28-10w%2B20%29\"$ Group the terms into two pairs.\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( " $\"w%28w-2%29%2B%28-10w%2B20%29\"$ Factor out the GCF $\"w\"$ from the first group.\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( " $\"w%28w-2%29-10%28w-2%29\"$ Factor out $\"10\"$ 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( " $\"%28w-10%29%28w-2%29\"$ Combine like terms. Or factor out the common term $\"w-2\"$ \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 $\"w%5E2-12w%2B20\"$ factors to $\"%28w-10%29%28w-2%29\"$ .\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "Note: you can check the answer by FOILing $\"%28w-10%29%28w-2%29\"$ to get $\"w%5E2-12w%2B20\"$ or by graphing the original expression and the answer (the two graphs should be identical).
\n" ); document.write( " \n" ); document.write( "

\n" );