document.write( "Question 507323: 36y^2+60y+25
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Algebra.Com's Answer #340578 by jim_thompson5910(35256) $\"\"$ $\"About$

You can put this solution on YOUR website!
I'm assuming you want to factor this.\r
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\n" ); document.write( "\n" ); document.write( "Looking at the expression $\"36y%5E2%2B60y%2B25\"$ , we can see that the first coefficient is $\"36\"$ , the second coefficient is $\"60\"$ , and the last term is $\"25\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Now multiply the first coefficient $\"36\"$ by the last term $\"25\"$ to get $\"%2836%29%2825%29=900\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Now the question is: what two whole numbers multiply to $\"900\"$ (the previous product) and add to the second coefficient $\"60\"$ ?\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 $\"900\"$ (the previous product).\r
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\n" ); document.write( "\n" ); document.write( "Factors of $\"900\"$ :\r
\n" ); document.write( "\n" ); document.write( "1,2,3,4,5,6,9,10,12,15,18,20,25,30,36,45,50,60,75,90,100,150,180,225,300,450,900\r
\n" ); document.write( "\n" ); document.write( "-1,-2,-3,-4,-5,-6,-9,-10,-12,-15,-18,-20,-25,-30,-36,-45,-50,-60,-75,-90,-100,-150,-180,-225,-300,-450,-900\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 $\"900\"$ .\r
\n" ); document.write( "\n" ); document.write( "1*900 = 900
\n" ); document.write( "2*450 = 900
\n" ); document.write( "3*300 = 900
\n" ); document.write( "4*225 = 900
\n" ); document.write( "5*180 = 900
\n" ); document.write( "6*150 = 900
\n" ); document.write( "9*100 = 900
\n" ); document.write( "10*90 = 900
\n" ); document.write( "12*75 = 900
\n" ); document.write( "15*60 = 900
\n" ); document.write( "18*50 = 900
\n" ); document.write( "20*45 = 900
\n" ); document.write( "25*36 = 900
\n" ); document.write( "30*30 = 900
\n" ); document.write( "(-1)*(-900) = 900
\n" ); document.write( "(-2)*(-450) = 900
\n" ); document.write( "(-3)*(-300) = 900
\n" ); document.write( "(-4)*(-225) = 900
\n" ); document.write( "(-5)*(-180) = 900
\n" ); document.write( "(-6)*(-150) = 900
\n" ); document.write( "(-9)*(-100) = 900
\n" ); document.write( "(-10)*(-90) = 900
\n" ); document.write( "(-12)*(-75) = 900
\n" ); document.write( "(-15)*(-60) = 900
\n" ); document.write( "(-18)*(-50) = 900
\n" ); document.write( "(-20)*(-45) = 900
\n" ); document.write( "(-25)*(-36) = 900
\n" ); document.write( "(-30)*(-30) = 900\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 $\"60\"$ :\r
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First Number	Second Number	Sum
1	900	1+900=901
2	450	2+450=452
3	300	3+300=303
4	225	4+225=229
5	180	5+180=185
6	150	6+150=156
9	100	9+100=109
10	90	10+90=100
12	75	12+75=87
15	60	15+60=75
18	50	18+50=68
20	45	20+45=65
25	36	25+36=61
30	30	30+30=60
-1	-900	-1+(-900)=-901
-2	-450	-2+(-450)=-452
-3	-300	-3+(-300)=-303
-4	-225	-4+(-225)=-229
-5	-180	-5+(-180)=-185
-6	-150	-6+(-150)=-156
-9	-100	-9+(-100)=-109
-10	-90	-10+(-90)=-100
-12	-75	-12+(-75)=-87
-15	-60	-15+(-60)=-75
-18	-50	-18+(-50)=-68
-20	-45	-20+(-45)=-65
-25	-36	-25+(-36)=-61
-30	-30	-30+(-30)=-60

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\n" ); document.write( "\n" ); document.write( "From the table, we can see that the two numbers $\"30\"$ and $\"30\"$ add to $\"60\"$ (the middle coefficient).\r
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\n" ); document.write( "\n" ); document.write( "So the two numbers $\"30\"$ and $\"30\"$ both multiply to $\"900\"$ and add to $\"60\"$ \r
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\n" ); document.write( "\n" ); document.write( "Now replace the middle term $\"60y\"$ with $\"30y%2B30y\"$ . Remember, $\"30\"$ and $\"30\"$ add to $\"60\"$ . So this shows us that $\"30y%2B30y=60y\"$ .\r
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\n" ); document.write( "\n" ); document.write( " $\"36y%5E2%2Bhighlight%2830y%2B30y%29%2B25\"$ Replace the second term $\"60y\"$ with $\"30y%2B30y\"$ .\r
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\n" ); document.write( "\n" ); document.write( " $\"%2836y%5E2%2B30y%29%2B%2830y%2B25%29\"$ Group the terms into two pairs.\r
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\n" ); document.write( "\n" ); document.write( " $\"6y%286y%2B5%29%2B%2830y%2B25%29\"$ Factor out the GCF $\"6y\"$ from the first group.\r
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\n" ); document.write( "\n" ); document.write( " $\"6y%286y%2B5%29%2B5%286y%2B5%29\"$ 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( " $\"%286y%2B5%29%286y%2B5%29\"$ Combine like terms. Or factor out the common term $\"6y%2B5\"$ \r
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\n" ); document.write( "\n" ); document.write( " $\"%286y%2B5%29%5E2\"$ Condense the terms.\r
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\n" ); document.write( "\n" ); document.write( "Answer:\r
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\n" ); document.write( "\n" ); document.write( "So $\"36y%5E2%2B60y%2B25\"$ factors to $\"%286y%2B5%29%5E2\"$ .\r
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\n" ); document.write( "\n" ); document.write( "In other words, $\"36y%5E2%2B60y%2B25=%286y%2B5%29%5E2\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Note: you can check the answer by expanding $\"%286y%2B5%29%5E2\"$ to get $\"36y%5E2%2B60y%2B25\"$ 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( "Let me know if you need more help or if you need me to explain a step in more detail.
\n" ); document.write( "Feel free to email me at jim_thompson5910@hotmail.com
\n" ); document.write( "or you can visit my website here: http://www.freewebs.com/jimthompson5910/home.html\r
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\n" ); document.write( "\n" ); document.write( "Thanks,\r
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\n" ); document.write( "\n" ); document.write( "Jim \n" ); document.write( "

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