document.write( "Question 189830: Factor completely\r
\n" ); document.write( "\n" ); document.write( "6a^2 – 6a - 72
\n" ); document.write( " \n" ); document.write( "

Algebra.Com's Answer #142428 by jim_thompson5910(35256) $\"\"$ $\"About$

You can put this solution on YOUR website!
$\"6a%5E2-6a-72\"$ Start with the given expression\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( " $\"6%28a%5E2-a-12%29\"$ Factor out the GCF $\"6\"$ \r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "Now let's focus on the inner expression $\"a%5E2-a-12\"$ \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 $\"a%5E2-a-12\"$ we can see that the first term is $\"a%5E2\"$ and the last term is $\"-12\"$ where the coefficients are 1 and -12 respectively.\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "Now multiply the first coefficient 1 and the last coefficient -12 to get -12. Now what two numbers multiply to -12 and add to the middle coefficient -1? Let's list all of the factors of -12:\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "Factors of -12:\r
\n" ); document.write( "\n" ); document.write( "1,2,3,4,6,12\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "-1,-2,-3,-4,-6,-12 ...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 -12\r
\n" ); document.write( "\n" ); document.write( "(1)*(-12)\r
\n" ); document.write( "\n" ); document.write( "(2)*(-6)\r
\n" ); document.write( "\n" ); document.write( "(3)*(-4)\r
\n" ); document.write( "\n" ); document.write( "(-1)*(12)\r
\n" ); document.write( "\n" ); document.write( "(-2)*(6)\r
\n" ); document.write( "\n" ); document.write( "(-3)*(4)\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "note: remember, the product of a negative and a positive number is a negative number\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "Now which of these pairs add to -1? Lets make a table of all of the pairs of factors we multiplied and see which two numbers add to -1\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "\n" ); document.write( "\n" ); document.write( "

First Number	Second Number	Sum
1	-12	1+(-12)=-11
2	-6	2+(-6)=-4
3	-4	3+(-4)=-1
-1	12	-1+12=11
-2	6	-2+6=4
-3	4	-3+4=1

\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "From this list we can see that 3 and -4 add up to -1 and multiply to -12\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "Now looking at the expression $\"a%5E2-a-12\"$ , replace $\"-a\"$ with $\"3a-4a\"$ (notice $\"3a-4a\"$ combines to $\"-a\"$ . So it is equivalent to $\"-a\"$ )\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( " $\"a%5E2%2Bhighlight%283a-4a%29-12\"$ \r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "Now let's factor $\"a%5E2%2B3a-4a-12\"$ by grouping:\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( " $\"%28a%5E2%2B3a%29%2B%28-4a-12%29\"$ Group like terms\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( " $\"a%28a%2B3%29-4%28a%2B3%29\"$ Factor out the GCF of $\"a\"$ out of the first group. Factor out the GCF of $\"-4\"$ out of the second group\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( " $\"%28a-4%29%28a%2B3%29\"$ Since we have a common term of $\"a%2B3\"$ , we can combine like terms\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "So $\"a%5E2%2B3a-4a-12\"$ factors to $\"%28a-4%29%28a%2B3%29\"$ \r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "So this also means that $\"a%5E2-a-12\"$ factors to $\"%28a-4%29%28a%2B3%29\"$ (since $\"a%5E2-a-12\"$ is equivalent to $\"a%5E2%2B3a-4a-12\"$ )\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 $\"6%28a%5E2-a-12%29\"$ factors to $\"6%28a-4%29%28a%2B3%29\"$ \r
\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 $\"6a%5E2-6a-72\"$ completely factors to $\"6%28a-4%29%28a%2B3%29\"$
\n" ); document.write( " \n" ); document.write( "

\n" );