document.write( "Question 168141: 2x^2-4x+2 \n" ); document.write( "

Algebra.Com's Answer #123954 by jim_thompson5910(35256) $\"\"$ $\"About$

You can put this solution on YOUR website!
\r
\n" ); document.write( "\n" ); document.write( "Looking at the expression $\"2x%5E2-4x%2B2\"$ , we can see that the first coefficient is $\"2\"$ , the second coefficient is $\"-4\"$ , and the last term is $\"2\"$ .\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "Now multiply the first coefficient $\"2\"$ by the last term $\"2\"$ to get $\"%282%29%282%29=4\"$ .\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "Now the question is: what two whole numbers multiply to $\"4\"$ (the previous product) and add to the second coefficient $\"-4\"$ ?\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 $\"4\"$ (the previous product).\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "Factors of $\"4\"$ :\r
\n" ); document.write( "\n" ); document.write( "1,2,4\r
\n" ); document.write( "\n" ); document.write( "-1,-2,-4\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 $\"4\"$ .\r
\n" ); document.write( "\n" ); document.write( "1*4
\n" ); document.write( "2*2
\n" ); document.write( "(-1)*(-4)
\n" ); document.write( "(-2)*(-2)\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 $\"-4\"$ :\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	4	1+4=5
2	2	2+2=4
-1	-4	-1+(-4)=-5
-2	-2	-2+(-2)=-4

\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 $\"-2\"$ add to $\"-4\"$ (the middle coefficient).\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "So the two numbers $\"-2\"$ and $\"-2\"$ both multiply to $\"4\"$ and add to $\"-4\"$ \r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "Now replace the middle term $\"-4x\"$ with $\"-2x-2x\"$ . Remember, $\"-2\"$ and $\"-2\"$ add to $\"-4\"$ . So this shows us that $\"-2x-2x=-4x\"$ .\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( " $\"2x%5E2%2Bhighlight%28-2x-2x%29%2B2\"$ Replace the second term $\"-4x\"$ with $\"-2x-2x\"$ .\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( " $\"%282x%5E2-2x%29%2B%28-2x%2B2%29\"$ Group the terms into two pairs.\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( " $\"2x%28x-1%29%2B%28-2x%2B2%29\"$ Factor out the GCF $\"2x\"$ from the first group.\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( " $\"2x%28x-1%29-2%28x-1%29\"$ Factor out $\"2\"$ 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( " $\"%282x-2%29%28x-1%29\"$ Combine like terms. Or factor out the common term $\"x-1\"$ \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 $\"2x%5E2-4x%2B2\"$ factors to $\"%282x-2%29%28x-1%29\"$ .\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "Note: you can check the answer by FOILing $\"%282x-2%29%28x-1%29\"$ to get $\"2x%5E2-4x%2B2\"$ or by graphing the original expression and the answer (the two graphs should be identical).
\n" ); document.write( " \n" ); document.write( "

\n" );