document.write( "Question 189835: Factor completely\r
\n" ); document.write( "\n" ); document.write( " 3x^2 + 30x + 75 \n" ); document.write( "

Algebra.Com's Answer #142427 by jim_thompson5910(35256) $\"\"$ $\"About$

You can put this solution on YOUR website!
$\"3x%5E2%2B30x%2B75\"$ Start with the given expression\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( " $\"3%28x%5E2%2B10x%2B25%29\"$ Factor out the GCF $\"3\"$ \r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "Now let's focus on the inner expression $\"x%5E2%2B10x%2B25\"$ \r
\n" ); document.write( "
\n" ); document.write( "
\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( "\n" ); document.write( "Looking at $\"x%5E2%2B10x%2B25\"$ we can see that the first term is $\"x%5E2\"$ and the last term is $\"25\"$ where the coefficients are 1 and 25 respectively.\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "Now multiply the first coefficient 1 and the last coefficient 25 to get 25. Now what two numbers multiply to 25 and add to the middle coefficient 10? Let's list all of the factors of 25:\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "Factors of 25:\r
\n" ); document.write( "\n" ); document.write( "1,5\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "-1,-5 ...List the negative factors as well. This will allow us to find all possible combinations\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "These factors pair up and multiply to 25\r
\n" ); document.write( "\n" ); document.write( "1*25\r
\n" ); document.write( "\n" ); document.write( "5*5\r
\n" ); document.write( "\n" ); document.write( "(-1)*(-25)\r
\n" ); document.write( "\n" ); document.write( "(-5)*(-5)\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "note: remember two negative numbers multiplied together make a positive number\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "Now which of these pairs add to 10? Lets make a table of all of the pairs of factors we multiplied and see which two numbers add to 10\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "\n" ); document.write( "\n" ); document.write( "

First Number	Second Number	Sum
1	25	1+25=26
5	5	5+5=10
-1	-25	-1+(-25)=-26
-5	-5	-5+(-5)=-10

\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "From this list we can see that 5 and 5 add up to 10 and multiply to 25\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "Now looking at the expression $\"x%5E2%2B10x%2B25\"$ , replace $\"10x\"$ with $\"5x%2B5x\"$ (notice $\"5x%2B5x\"$ adds up to $\"10x\"$ . So it is equivalent to $\"10x\"$ )\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( " $\"x%5E2%2Bhighlight%285x%2B5x%29%2B25\"$ \r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "Now let's factor $\"x%5E2%2B5x%2B5x%2B25\"$ by grouping:\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( " $\"%28x%5E2%2B5x%29%2B%285x%2B25%29\"$ Group like terms\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( " $\"x%28x%2B5%29%2B5%28x%2B5%29\"$ Factor out the GCF of $\"x\"$ out of the first group. Factor out the GCF of $\"5\"$ out of the second group\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( " $\"%28x%2B5%29%28x%2B5%29\"$ Since we have a common term of $\"x%2B5\"$ , we can combine like terms\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "So $\"x%5E2%2B5x%2B5x%2B25\"$ factors to $\"%28x%2B5%29%28x%2B5%29\"$ \r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "So this also means that $\"x%5E2%2B10x%2B25\"$ factors to $\"%28x%2B5%29%28x%2B5%29\"$ (since $\"x%5E2%2B10x%2B25\"$ is equivalent to $\"x%5E2%2B5x%2B5x%2B25\"$ )\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "note: $\"%28x%2B5%29%28x%2B5%29\"$ is equivalent to $\"%28x%2B5%29%5E2\"$ since the term $\"x%2B5\"$ occurs twice. So $\"x%5E2%2B10x%2B25\"$ also factors to $\"%28x%2B5%29%5E2\"$ \r
\n" ); document.write( "
\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 $\"3%28x%5E2%2B10x%2B25%29\"$ factors down to $\"3%28x%2B5%29%5E2\"$ \r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "==============================================================\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "Answer:\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "So $\"3x%5E2%2B30x%2B75\"$ completely factors to $\"3%28x%2B5%29%5E2\"$ \n" ); document.write( "

\n" );