document.write( "Question 682271: 2x^2+6x+4 \n" ); document.write( "

Algebra.Com's Answer #423113 by jim_thompson5910(35256) $\"\"$ $\"About$

You can put this solution on YOUR website!
I'm assuming you want to factor this\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( " $\"2x%5E2%2B6x%2B4\"$ Start with the given expression.\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( " $\"2%28x%5E2%2B3x%2B2%29\"$ 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 $\"x%5E2%2B3x%2B2\"$ \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 $\"x%5E2%2B3x%2B2\"$ , we can see that the first coefficient is $\"1\"$ , the second coefficient is $\"3\"$ , and the last term is $\"2\"$ .\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "Now multiply the first coefficient $\"1\"$ by the last term $\"2\"$ to get $\"%281%29%282%29=2\"$ .\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "Now the question is: what two whole numbers multiply to $\"2\"$ (the previous product) and add to the second coefficient $\"3\"$ ?\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 $\"2\"$ (the previous product).\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "Factors of $\"2\"$ :\r
\n" ); document.write( "\n" ); document.write( "1,2\r
\n" ); document.write( "\n" ); document.write( "-1,-2\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 $\"2\"$ .\r
\n" ); document.write( "\n" ); document.write( "1*2 = 2
\n" ); document.write( "(-1)*(-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 $\"3\"$ :\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	2	1+2=3
-1	-2	-1+(-2)=-3

\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 $\"2\"$ add to $\"3\"$ (the middle coefficient).\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "So the two numbers $\"1\"$ and $\"2\"$ both multiply to $\"2\"$ and add to $\"3\"$ \r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "Now replace the middle term $\"3x\"$ with $\"x%2B2x\"$ . Remember, $\"1\"$ and $\"2\"$ add to $\"3\"$ . So this shows us that $\"x%2B2x=3x\"$ .\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( " $\"x%5E2%2Bhighlight%28x%2B2x%29%2B2\"$ Replace the second term $\"3x\"$ with $\"x%2B2x\"$ .\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( " $\"%28x%5E2%2Bx%29%2B%282x%2B2%29\"$ Group the terms into two pairs.\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( " $\"x%28x%2B1%29%2B%282x%2B2%29\"$ Factor out the GCF $\"x\"$ from the first group.\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( " $\"x%28x%2B1%29%2B2%28x%2B1%29\"$ 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( " $\"%28x%2B2%29%28x%2B1%29\"$ Combine like terms. Or factor out the common term $\"x%2B1\"$ \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 $\"2%28x%5E2%2B3x%2B2%29\"$ then factors further to $\"2%28x%2B2%29%28x%2B1%29\"$ \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 $\"2x%5E2%2B6x%2B4\"$ completely factors to $\"2%28x%2B2%29%28x%2B1%29\"$ .\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "In other words, $\"2x%5E2%2B6x%2B4=2%28x%2B2%29%28x%2B1%29\"$ .\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "Note: you can check the answer by expanding $\"2%28x%2B2%29%28x%2B1%29\"$ to get $\"2x%5E2%2B6x%2B4\"$ or by graphing the original expression and the answer (the two graphs should be identical).
\n" ); document.write( " \n" ); document.write( "

\n" );