document.write( "Question 551360: Factor Completely\r
\n" ); document.write( "\n" ); document.write( "3a^2+5ab-12b^2 \n" ); document.write( "

Algebra.Com's Answer #359677 by jim_thompson5910(35256) $\"\"$ $\"About$

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\n" ); document.write( "\n" ); document.write( "Looking at the expression $\"3a%5E2%2B5ab-12b%5E2\"$ , we can see that the first coefficient is $\"3\"$ , the second coefficient is $\"5\"$ , and the last coefficient is $\"-12\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Now multiply the first coefficient $\"3\"$ by the last coefficient $\"-12\"$ to get $\"%283%29%28-12%29=-36\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Now the question is: what two whole numbers multiply to $\"-36\"$ (the previous product) and add to the second coefficient $\"5\"$ ?\r
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\n" ); document.write( "\n" ); document.write( "To find these two numbers, we need to list all of the factors of $\"-36\"$ (the previous product).\r
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\n" ); document.write( "\n" ); document.write( "Factors of $\"-36\"$ :\r
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\n" ); document.write( "\n" ); document.write( "-1,-2,-3,-4,-6,-9,-12,-18,-36\r
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\n" ); document.write( "\n" ); document.write( "Note: list the negative of each factor. 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 $\"-36\"$ .\r
\n" ); document.write( "\n" ); document.write( "1*(-36) = -36
\n" ); document.write( "2*(-18) = -36
\n" ); document.write( "3*(-12) = -36
\n" ); document.write( "4*(-9) = -36
\n" ); document.write( "6*(-6) = -36
\n" ); document.write( "(-1)*(36) = -36
\n" ); document.write( "(-2)*(18) = -36
\n" ); document.write( "(-3)*(12) = -36
\n" ); document.write( "(-4)*(9) = -36
\n" ); document.write( "(-6)*(6) = -36\r
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\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 $\"5\"$ :\r
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First Number	Second Number	Sum
1	-36	1+(-36)=-35
2	-18	2+(-18)=-16
3	-12	3+(-12)=-9
4	-9	4+(-9)=-5
6	-6	6+(-6)=0
-1	36	-1+36=35
-2	18	-2+18=16
-3	12	-3+12=9
-4	9	-4+9=5
-6	6	-6+6=0

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\n" ); document.write( "\n" ); document.write( "From the table, we can see that the two numbers $\"-4\"$ and $\"9\"$ add to $\"5\"$ (the middle coefficient).\r
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\n" ); document.write( "\n" ); document.write( "So the two numbers $\"-4\"$ and $\"9\"$ both multiply to $\"-36\"$ and add to $\"5\"$ \r
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\n" ); document.write( "\n" ); document.write( "Now replace the middle term $\"5ab\"$ with $\"-4ab%2B9ab\"$ . Remember, $\"-4\"$ and $\"9\"$ add to $\"5\"$ . So this shows us that $\"-4ab%2B9ab=5ab\"$ .\r
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\n" ); document.write( "\n" ); document.write( " $\"3a%5E2%2Bhighlight%28-4ab%2B9ab%29-12b%5E2\"$ Replace the second term $\"5ab\"$ with $\"-4ab%2B9ab\"$ .\r
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\n" ); document.write( "\n" ); document.write( " $\"%283a%5E2-4ab%29%2B%289ab-12b%5E2%29\"$ Group the terms into two pairs.\r
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\n" ); document.write( "\n" ); document.write( " $\"a%283a-4b%29%2B%289ab-12b%5E2%29\"$ Factor out the GCF $\"a\"$ from the first group.\r
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\n" ); document.write( "\n" ); document.write( " $\"a%283a-4b%29%2B3b%283a-4b%29\"$ Factor out $\"3b\"$ 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
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\n" ); document.write( "\n" ); document.write( " $\"%28a%2B3b%29%283a-4b%29\"$ Combine like terms. Or factor out the common term $\"3a-4b\"$ \r
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\n" ); document.write( "\n" ); document.write( "Answer:\r
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\n" ); document.write( "\n" ); document.write( "So $\"3a%5E2%2B5ab-12b%5E2\"$ factors to $\"%28a%2B3b%29%283a-4b%29\"$ .\r
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\n" ); document.write( "\n" ); document.write( "In other words, $\"3a%5E2%2B5ab-12b%5E2=%28a%2B3b%29%283a-4b%29\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Note: you can check the answer by expanding $\"%28a%2B3b%29%283a-4b%29\"$ to get $\"3a%5E2%2B5ab-12b%5E2\"$ or by graphing the original expression and the answer (the two graphs should be identical).
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