document.write( "Question 201025: factor completely
\n" ); document.write( "25m^2+81-90m \n" ); document.write( "

Algebra.Com's Answer #151263 by jim_thompson5910(35256) $\"\"$ $\"About$

You can put this solution on YOUR website!
$\"25m%5E2%2B81-90m\"$ Start with the given expression.\r
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\n" ); document.write( "\n" ); document.write( " $\"25m%5E2-90m%2B81\"$ Rearrange the terms in descending exponential order.\r
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\n" ); document.write( "\n" ); document.write( "Looking at the expression $\"25m%5E2-90m%2B81\"$ , we can see that the first coefficient is $\"25\"$ , the second coefficient is $\"-90\"$ , and the last term is $\"81\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Now multiply the first coefficient $\"25\"$ by the last term $\"81\"$ to get $\"%2825%29%2881%29=2025\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Now the question is: what two whole numbers multiply to $\"2025\"$ (the previous product) and add to the second coefficient $\"-90\"$ ?\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 $\"2025\"$ (the previous product).\r
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\n" ); document.write( "\n" ); document.write( "Factors of $\"2025\"$ :\r
\n" ); document.write( "\n" ); document.write( "1,3,5,9,15,25,27,45,75,81,135,225,405,675,2025\r
\n" ); document.write( "\n" ); document.write( "-1,-3,-5,-9,-15,-25,-27,-45,-75,-81,-135,-225,-405,-675,-2025\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 $\"2025\"$ .\r
\n" ); document.write( "\n" ); document.write( "1*2025
\n" ); document.write( "3*675
\n" ); document.write( "5*405
\n" ); document.write( "9*225
\n" ); document.write( "15*135
\n" ); document.write( "25*81
\n" ); document.write( "27*75
\n" ); document.write( "45*45
\n" ); document.write( "(-1)*(-2025)
\n" ); document.write( "(-3)*(-675)
\n" ); document.write( "(-5)*(-405)
\n" ); document.write( "(-9)*(-225)
\n" ); document.write( "(-15)*(-135)
\n" ); document.write( "(-25)*(-81)
\n" ); document.write( "(-27)*(-75)
\n" ); document.write( "(-45)*(-45)\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 $\"-90\"$ :\r
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First Number	Second Number	Sum
1	2025	1+2025=2026
3	675	3+675=678
5	405	5+405=410
9	225	9+225=234
15	135	15+135=150
25	81	25+81=106
27	75	27+75=102
45	45	45+45=90
-1	-2025	-1+(-2025)=-2026
-3	-675	-3+(-675)=-678
-5	-405	-5+(-405)=-410
-9	-225	-9+(-225)=-234
-15	-135	-15+(-135)=-150
-25	-81	-25+(-81)=-106
-27	-75	-27+(-75)=-102
-45	-45	-45+(-45)=-90

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\n" ); document.write( "\n" ); document.write( "From the table, we can see that the two numbers $\"-45\"$ and $\"-45\"$ add to $\"-90\"$ (the middle coefficient).\r
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\n" ); document.write( "\n" ); document.write( "So the two numbers $\"-45\"$ and $\"-45\"$ both multiply to $\"2025\"$ and add to $\"-90\"$ \r
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\n" ); document.write( "\n" ); document.write( "Now replace the middle term $\"-90m\"$ with $\"-45m-45m\"$ . Remember, $\"-45\"$ and $\"-45\"$ add to $\"-90\"$ . So this shows us that $\"-45m-45m=-90m\"$ .\r
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\n" ); document.write( "\n" ); document.write( " $\"25m%5E2%2Bhighlight%28-45m-45m%29%2B81\"$ Replace the second term $\"-90m\"$ with $\"-45m-45m\"$ .\r
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\n" ); document.write( "\n" ); document.write( " $\"%2825m%5E2-45m%29%2B%28-45m%2B81%29\"$ Group the terms into two pairs.\r
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\n" ); document.write( "\n" ); document.write( " $\"5m%285m-9%29%2B%28-45m%2B81%29\"$ Factor out the GCF $\"5m\"$ from the first group.\r
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\n" ); document.write( "\n" ); document.write( " $\"5m%285m-9%29-9%285m-9%29\"$ Factor out $\"9\"$ 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( " $\"%285m-9%29%285m-9%29\"$ Combine like terms. Or factor out the common term $\"5m-9\"$ \r
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\n" ); document.write( "\n" ); document.write( " $\"%285m-9%29%5E2\"$ Condense\r
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\n" ); document.write( "\n" ); document.write( "Answer:\r
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\n" ); document.write( "\n" ); document.write( "So $\"25m%5E2%2B81-90m\"$ factors to $\"%285m-9%29%5E2\"$ .\r
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\n" ); document.write( "\n" ); document.write( "In other words, $\"25m%5E2%2B81-90m=%285m-9%29%5E2\"$ \r
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\n" ); document.write( "\n" ); document.write( "Note: you can check the answer by FOILing $\"%285m-9%29%5E2\"$ to get $\"25m%5E2%2B81-90m\"$ or by graphing the original expression and the answer (the two graphs should be identical).
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