document.write( "Question 387339: Factor completly\r
\n" ); document.write( "\n" ); document.write( "(20b^2+100bt+125t^2)\r
\n" ); document.write( "\n" ); document.write( "I understand the GCF is 5 and I had tried to factor it out getting
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\n" ); document.write( "\n" ); document.write( "No matter where I place the numbers or what numbers I use I can't get both 100bt AND 125t^2. I e-mailed my teacher for help 5 days ago with still no response and at this point any help is REALLY appreciated!!!
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Algebra.Com's Answer #273791 by jim_thompson5910(35256) $\"\"$ $\"About$

You can put this solution on YOUR website!
$\"20b%5E2%2B100bt%2B125t%5E2\"$ Start with the given expression.\r
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\n" ); document.write( "\n" ); document.write( " $\"5%284b%5E2%2B20bt%2B25t%5E2%29\"$ Factor out the GCF $\"5\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Now let's try to factor the inner expression $\"4b%5E2%2B20bt%2B25t%5E2\"$ \r
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\n" ); document.write( "\n" ); document.write( "Looking at the expression $\"4b%5E2%2B20bt%2B25t%5E2\"$ , we can see that the first coefficient is $\"4\"$ , the second coefficient is $\"20\"$ , and the last coefficient is $\"25\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Now multiply the first coefficient $\"4\"$ by the last coefficient $\"25\"$ to get $\"%284%29%2825%29=100\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Now the question is: what two whole numbers multiply to $\"100\"$ (the previous product) and add to the second coefficient $\"20\"$ ?\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 $\"100\"$ (the previous product).\r
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\n" ); document.write( "\n" ); document.write( "Factors of $\"100\"$ :\r
\n" ); document.write( "\n" ); document.write( "1,2,4,5,10,20,25,50,100\r
\n" ); document.write( "\n" ); document.write( "-1,-2,-4,-5,-10,-20,-25,-50,-100\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 $\"100\"$ .\r
\n" ); document.write( "\n" ); document.write( "1*100 = 100
\n" ); document.write( "2*50 = 100
\n" ); document.write( "4*25 = 100
\n" ); document.write( "5*20 = 100
\n" ); document.write( "10*10 = 100
\n" ); document.write( "(-1)*(-100) = 100
\n" ); document.write( "(-2)*(-50) = 100
\n" ); document.write( "(-4)*(-25) = 100
\n" ); document.write( "(-5)*(-20) = 100
\n" ); document.write( "(-10)*(-10) = 100\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 $\"20\"$ :\r
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First Number	Second Number	Sum
1	100	1+100=101
2	50	2+50=52
4	25	4+25=29
5	20	5+20=25
10	10	10+10=20
-1	-100	-1+(-100)=-101
-2	-50	-2+(-50)=-52
-4	-25	-4+(-25)=-29
-5	-20	-5+(-20)=-25
-10	-10	-10+(-10)=-20

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\n" ); document.write( "\n" ); document.write( "From the table, we can see that the two numbers $\"10\"$ and $\"10\"$ add to $\"20\"$ (the middle coefficient).\r
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\n" ); document.write( "\n" ); document.write( "So the two numbers $\"10\"$ and $\"10\"$ both multiply to $\"100\"$ and add to $\"20\"$ \r
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\n" ); document.write( "\n" ); document.write( "Now replace the middle term $\"20bt\"$ with $\"10bt%2B10bt\"$ . Remember, $\"10\"$ and $\"10\"$ add to $\"20\"$ . So this shows us that $\"10bt%2B10bt=20bt\"$ .\r
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\n" ); document.write( "\n" ); document.write( " $\"4b%5E2%2Bhighlight%2810bt%2B10bt%29%2B25t%5E2\"$ Replace the second term $\"20bt\"$ with $\"10bt%2B10bt\"$ .\r
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\n" ); document.write( "\n" ); document.write( " $\"%284b%5E2%2B10bt%29%2B%2810bt%2B25t%5E2%29\"$ Group the terms into two pairs.\r
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\n" ); document.write( "\n" ); document.write( " $\"2b%282b%2B5t%29%2B%2810bt%2B25t%5E2%29\"$ Factor out the GCF $\"2b\"$ from the first group.\r
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\n" ); document.write( "\n" ); document.write( " $\"2b%282b%2B5t%29%2B5t%282b%2B5t%29\"$ Factor out $\"5t\"$ 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( " $\"%282b%2B5t%29%282b%2B5t%29\"$ Combine like terms. Or factor out the common term $\"2b%2B5t\"$ \r
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\n" ); document.write( "\n" ); document.write( " $\"%282b%2B5t%29%5E2\"$ Condense the terms.\r
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\n" ); document.write( "\n" ); document.write( "So $\"5%284b%5E2%2B20bt%2B25t%5E2%29\"$ then factors further to $\"5%282b%2B5t%29%5E2\"$ \r
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\n" ); document.write( "\n" ); document.write( "Answer:\r
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\n" ); document.write( "\n" ); document.write( "So $\"20b%5E2%2B100bt%2B125t%5E2\"$ completely factors to $\"5%282b%2B5t%29%5E2\"$ .\r
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\n" ); document.write( "\n" ); document.write( "In other words, $\"20b%5E2%2B100bt%2B125t%5E2=5%282b%2B5t%29%5E2\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Note: you can check the answer by expanding $\"5%282b%2B5t%29%5E2\"$ to get $\"20b%5E2%2B100bt%2B125t%5E2\"$ or by graphing the original expression and the answer (the two graphs should be identical).\r
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\n" ); document.write( "\n" ); document.write( "If you need more help, email me at jim_thompson5910@hotmail.com\r
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\n" ); document.write( "\n" ); document.write( "Also, feel free to check out my tutoring website\r
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\n" ); document.write( "\n" ); document.write( "Jim \n" ); document.write( "

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