document.write( "Question 299484: reverse foil or trial factoring\r
\n" ); document.write( "\n" ); document.write( "9x^2-25 \n" ); document.write( "

Algebra.Com's Answer #215168 by jim_thompson5910(35256) $\"\"$ $\"About$

You can put this solution on YOUR website!
Note: $\"9x%5E2-25\"$ can be written as $\"9x%5E2%2B0x-25\"$ \r
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\n" ); document.write( "\n" ); document.write( "Looking at the expression $\"9x%5E2%2B0x-25\"$ , we can see that the first coefficient is $\"9\"$ , the second coefficient is $\"0\"$ , and the last term is $\"-25\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Now multiply the first coefficient $\"9\"$ by the last term $\"-25\"$ to get $\"%289%29%28-25%29=-225\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Now the question is: what two whole numbers multiply to $\"-225\"$ (the previous product) and add to the second coefficient $\"0\"$ ?\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 $\"-225\"$ (the previous product).\r
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\n" ); document.write( "\n" ); document.write( "Factors of $\"-225\"$ :\r
\n" ); document.write( "\n" ); document.write( "1,3,5,9,15,25,45,75,225\r
\n" ); document.write( "\n" ); document.write( "-1,-3,-5,-9,-15,-25,-45,-75,-225\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 $\"-225\"$ .\r
\n" ); document.write( "\n" ); document.write( "1*(-225) = -225
\n" ); document.write( "3*(-75) = -225
\n" ); document.write( "5*(-45) = -225
\n" ); document.write( "9*(-25) = -225
\n" ); document.write( "15*(-15) = -225
\n" ); document.write( "(-1)*(225) = -225
\n" ); document.write( "(-3)*(75) = -225
\n" ); document.write( "(-5)*(45) = -225
\n" ); document.write( "(-9)*(25) = -225
\n" ); document.write( "(-15)*(15) = -225\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 $\"0\"$ :\r
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First Number	Second Number	Sum
1	-225	1+(-225)=-224
3	-75	3+(-75)=-72
5	-45	5+(-45)=-40
9	-25	9+(-25)=-16
15	-15	15+(-15)=0
-1	225	-1+225=224
-3	75	-3+75=72
-5	45	-5+45=40
-9	25	-9+25=16
-15	15	-15+15=0

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\n" ); document.write( "\n" ); document.write( "From the table, we can see that the two numbers $\"-15\"$ and $\"15\"$ add to $\"0\"$ (the middle coefficient).\r
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\n" ); document.write( "\n" ); document.write( "So the two numbers $\"-15\"$ and $\"15\"$ both multiply to $\"-225\"$ and add to $\"0\"$ \r
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\n" ); document.write( "\n" ); document.write( "Now replace the middle term $\"0x\"$ with $\"-15x%2B15x\"$ . Remember, $\"-15\"$ and $\"15\"$ add to $\"0\"$ . So this shows us that $\"-15x%2B15x=0x\"$ .\r
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\n" ); document.write( "\n" ); document.write( " $\"9x%5E2%2Bhighlight%28-15x%2B15x%29-25\"$ Replace the second term $\"0x\"$ with $\"-15x%2B15x\"$ .\r
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\n" ); document.write( "\n" ); document.write( " $\"%289x%5E2-15x%29%2B%2815x-25%29\"$ Group the terms into two pairs.\r
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\n" ); document.write( "\n" ); document.write( " $\"3x%283x-5%29%2B%2815x-25%29\"$ Factor out the GCF $\"3x\"$ from the first group.\r
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\n" ); document.write( "\n" ); document.write( " $\"3x%283x-5%29%2B5%283x-5%29\"$ Factor out $\"5\"$ 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( " $\"%283x%2B5%29%283x-5%29\"$ Combine like terms. Or factor out the common term $\"3x-5\"$ \r
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\n" ); document.write( "\n" ); document.write( "Answer:\r
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\n" ); document.write( "\n" ); document.write( "So $\"9x%5E2%2B0x-25\"$ factors to $\"%283x%2B5%29%283x-5%29\"$ .\r
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\n" ); document.write( "\n" ); document.write( "In other words, $\"9x%5E2%2B0x-25=%283x%2B5%29%283x-5%29\"$ .\r
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\n" ); document.write( "\n" ); document.write( "So $\"9x%5E2-25=%283x%2B5%29%283x-5%29\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Note: you can check the answer by expanding $\"%283x%2B5%29%283x-5%29\"$ to get $\"9x%5E2%2B0x-25\"$ or by graphing the original expression and the answer (the two graphs should be identical). Also, $\"9x%5E2-25\"$ is a difference of squares.
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