document.write( "Question 357489: factor completley p2+2pq-24q2 \n" ); document.write( "

Algebra.Com's Answer #255257 by jim_thompson5910(35256) $\"\"$ $\"About$

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\n" ); document.write( "\n" ); document.write( "Looking at $\"p%5E2%2B2pq-24q%5E2\"$ we can see that the first term is $\"p%5E2\"$ and the last term is $\"-24q%5E2\"$ where the coefficients are 1 and -24 respectively.\r
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\n" ); document.write( "\n" ); document.write( "Now multiply the first coefficient 1 and the last coefficient -24 to get -24. Now what two numbers multiply to -24 and add to the middle coefficient 2? Let's list all of the factors of -24:\r
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\n" ); document.write( "\n" ); document.write( "Factors of -24:\r
\n" ); document.write( "\n" ); document.write( "1,2,3,4,6,8,12,24\r
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\n" ); document.write( "\n" ); document.write( "-1,-2,-3,-4,-6,-8,-12,-24 ...List the negative factors as well. This will allow us to find all possible combinations\r
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\n" ); document.write( "\n" ); document.write( "These factors pair up and multiply to -24\r
\n" ); document.write( "\n" ); document.write( "(1)*(-24)\r
\n" ); document.write( "\n" ); document.write( "(2)*(-12)\r
\n" ); document.write( "\n" ); document.write( "(3)*(-8)\r
\n" ); document.write( "\n" ); document.write( "(4)*(-6)\r
\n" ); document.write( "\n" ); document.write( "(-1)*(24)\r
\n" ); document.write( "\n" ); document.write( "(-2)*(12)\r
\n" ); document.write( "\n" ); document.write( "(-3)*(8)\r
\n" ); document.write( "\n" ); document.write( "(-4)*(6)\r
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\n" ); document.write( "\n" ); document.write( "note: remember, the product of a negative and a positive number is a negative number\r
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\n" ); document.write( "\n" ); document.write( "Now which of these pairs add to 2? Lets make a table of all of the pairs of factors we multiplied and see which two numbers add to 2\r
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First Number	Second Number	Sum
1	-24	1+(-24)=-23
2	-12	2+(-12)=-10
3	-8	3+(-8)=-5
4	-6	4+(-6)=-2
-1	24	-1+24=23
-2	12	-2+12=10
-3	8	-3+8=5
-4	6	-4+6=2

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\n" ); document.write( "\n" ); document.write( "From this list we can see that -4 and 6 add up to 2 and multiply to -24\r
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\n" ); document.write( "\n" ); document.write( "Now looking at the expression $\"p%5E2%2B2pq-24q%5E2\"$ , replace $\"2pq\"$ with $\"-4pq%2B6pq\"$ (notice $\"-4pq%2B6pq\"$ adds up to $\"2pq\"$ . So it is equivalent to $\"2pq\"$ )\r
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\n" ); document.write( "\n" ); document.write( " $\"p%5E2%2Bhighlight%28-4pq%2B6pq%29%2B-24q%5E2\"$ \r
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\n" ); document.write( "\n" ); document.write( "Now let's factor $\"p%5E2-4pq%2B6pq-24q%5E2\"$ by grouping:\r
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\n" ); document.write( "\n" ); document.write( " $\"%28p%5E2-4pq%29%2B%286pq-24q%5E2%29\"$ Group like terms\r
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\n" ); document.write( "\n" ); document.write( " $\"p%28p-4q%29%2B6q%28p-4q%29\"$ Factor out the GCF of $\"p\"$ out of the first group. Factor out the GCF of $\"6q\"$ out of the second group\r
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\n" ); document.write( "\n" ); document.write( " $\"%28p%2B6q%29%28p-4q%29\"$ Since we have a common term of $\"p-4q\"$ , we can combine like terms\r
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\n" ); document.write( "\n" ); document.write( "So $\"p%5E2-4pq%2B6pq-24q%5E2\"$ factors to $\"%28p%2B6q%29%28p-4q%29\"$ \r
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\n" ); document.write( "\n" ); document.write( "So this also means that $\"p%5E2%2B2pq-24q%5E2\"$ factors to $\"%28p%2B6q%29%28p-4q%29\"$ (since $\"p%5E2%2B2pq-24q%5E2\"$ is equivalent to $\"p%5E2-4pq%2B6pq-24q%5E2\"$ )\r
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\n" ); document.write( "\n" ); document.write( " Answer:\r
\n" ); document.write( "\n" ); document.write( "So $\"p%5E2%2B2pq-24q%5E2\"$ factors to $\"%28p%2B6q%29%28p-4q%29\"$
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