document.write( "Question 318368: Factor completely:\r
\n" ); document.write( "\n" ); document.write( "8p^2 + 7pq - 18q^2 \r
\n" ); document.write( "\n" ); document.write( "Can someone please explain to me in detail how to answer this question.. \n" ); document.write( "

Algebra.Com's Answer #227878 by jim_thompson5910(35256) $\"\"$ $\"About$

You can put this solution on YOUR website!
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\n" ); document.write( "\n" ); document.write( "Looking at $\"8p%5E2%2B7pq-18q%5E2\"$ we can see that the first term is $\"8p%5E2\"$ and the last term is $\"-18q%5E2\"$ where the coefficients are 8 and -18 respectively.\r
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\n" ); document.write( "\n" ); document.write( "Now multiply the first coefficient 8 and the last coefficient -18 to get -144. Now what two numbers multiply to -144 and add to the middle coefficient 7? Let's list all of the factors of -144:\r
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\n" ); document.write( "\n" ); document.write( "Factors of -144:\r
\n" ); document.write( "\n" ); document.write( "1,2,3,4,6,8,9,12,16,18,24,36,48,72\r
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\n" ); document.write( "\n" ); document.write( "-1,-2,-3,-4,-6,-8,-9,-12,-16,-18,-24,-36,-48,-72 ...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 -144\r
\n" ); document.write( "\n" ); document.write( "(1)*(-144)\r
\n" ); document.write( "\n" ); document.write( "(2)*(-72)\r
\n" ); document.write( "\n" ); document.write( "(3)*(-48)\r
\n" ); document.write( "\n" ); document.write( "(4)*(-36)\r
\n" ); document.write( "\n" ); document.write( "(6)*(-24)\r
\n" ); document.write( "\n" ); document.write( "(8)*(-18)\r
\n" ); document.write( "\n" ); document.write( "(9)*(-16)\r
\n" ); document.write( "\n" ); document.write( "(-1)*(144)\r
\n" ); document.write( "\n" ); document.write( "(-2)*(72)\r
\n" ); document.write( "\n" ); document.write( "(-3)*(48)\r
\n" ); document.write( "\n" ); document.write( "(-4)*(36)\r
\n" ); document.write( "\n" ); document.write( "(-6)*(24)\r
\n" ); document.write( "\n" ); document.write( "(-8)*(18)\r
\n" ); document.write( "\n" ); document.write( "(-9)*(16)\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 7? Lets make a table of all of the pairs of factors we multiplied and see which two numbers add to 7\r
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First Number	Second Number	Sum
1	-144	1+(-144)=-143
2	-72	2+(-72)=-70
3	-48	3+(-48)=-45
4	-36	4+(-36)=-32
6	-24	6+(-24)=-18
8	-18	8+(-18)=-10
9	-16	9+(-16)=-7
-1	144	-1+144=143
-2	72	-2+72=70
-3	48	-3+48=45
-4	36	-4+36=32
-6	24	-6+24=18
-8	18	-8+18=10
-9	16	-9+16=7

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\n" ); document.write( "\n" ); document.write( "From this list we can see that -9 and 16 add up to 7 and multiply to -144\r
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\n" ); document.write( "\n" ); document.write( "Now looking at the expression $\"8p%5E2%2B7pq-18q%5E2\"$ , replace $\"7pq\"$ with $\"-9pq%2B16pq\"$ (notice $\"-9pq%2B16pq\"$ adds up to $\"7pq\"$ . So it is equivalent to $\"7pq\"$ )\r
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\n" ); document.write( "\n" ); document.write( " $\"8p%5E2%2Bhighlight%28-9pq%2B16pq%29%2B-18q%5E2\"$ \r
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\n" ); document.write( "\n" ); document.write( "Now let's factor $\"8p%5E2-9pq%2B16pq-18q%5E2\"$ by grouping:\r
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\n" ); document.write( "\n" ); document.write( " $\"%288p%5E2-9pq%29%2B%2816pq-18q%5E2%29\"$ Group like terms\r
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\n" ); document.write( "\n" ); document.write( " $\"p%288p-9q%29%2B2q%288p-9q%29\"$ Factor out the GCF of $\"p\"$ out of the first group. Factor out the GCF of $\"2q\"$ out of the second group\r
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\n" ); document.write( "\n" ); document.write( " $\"%28p%2B2q%29%288p-9q%29\"$ Since we have a common term of $\"8p-9q\"$ , we can combine like terms\r
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\n" ); document.write( "\n" ); document.write( "So $\"8p%5E2-9pq%2B16pq-18q%5E2\"$ factors to $\"%28p%2B2q%29%288p-9q%29\"$ \r
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\n" ); document.write( "\n" ); document.write( "So this also means that $\"8p%5E2%2B7pq-18q%5E2\"$ factors to $\"%28p%2B2q%29%288p-9q%29\"$ (since $\"8p%5E2%2B7pq-18q%5E2\"$ is equivalent to $\"8p%5E2-9pq%2B16pq-18q%5E2\"$ )\r
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\n" ); document.write( "\n" ); document.write( " Answer:\r
\n" ); document.write( "\n" ); document.write( "So $\"8p%5E2%2B7pq-18q%5E2\"$ factors to $\"%28p%2B2q%29%288p-9q%29\"$
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