document.write( "Question 171351: 75b^2-180b+108 \n" ); document.write( "

Algebra.Com's Answer #126495 by jim_thompson5910(35256) $\"\"$ $\"About$

You can put this solution on YOUR website!
I'm assuming that you want to factor\r
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\n" ); document.write( "\n" ); document.write( " $\"75b%5E2-180b%2B108\"$ Start with the given expression\r
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\n" ); document.write( "\n" ); document.write( " $\"3%2825b%5E2-60b%2B36%29\"$ Factor out the GCF $\"3\"$ \r
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\n" ); document.write( "\n" ); document.write( "Now let's focus on the inner expression $\"25b%5E2-60b%2B36\"$ \r
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\n" ); document.write( "\n" ); document.write( "Looking at $\"25b%5E2-60b%2B36\"$ we can see that the first term is $\"25b%5E2\"$ and the last term is $\"36\"$ where the coefficients are 25 and 36 respectively.\r
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\n" ); document.write( "\n" ); document.write( "Now multiply the first coefficient 25 and the last coefficient 36 to get 900. Now what two numbers multiply to 900 and add to the middle coefficient -60? Let's list all of the factors of 900:\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\r
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\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 ...List the negative factors as well. 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\r
\n" ); document.write( "\n" ); document.write( "2*450\r
\n" ); document.write( "\n" ); document.write( "3*300\r
\n" ); document.write( "\n" ); document.write( "4*225\r
\n" ); document.write( "\n" ); document.write( "5*180\r
\n" ); document.write( "\n" ); document.write( "6*150\r
\n" ); document.write( "\n" ); document.write( "9*100\r
\n" ); document.write( "\n" ); document.write( "10*90\r
\n" ); document.write( "\n" ); document.write( "12*75\r
\n" ); document.write( "\n" ); document.write( "15*60\r
\n" ); document.write( "\n" ); document.write( "18*50\r
\n" ); document.write( "\n" ); document.write( "20*45\r
\n" ); document.write( "\n" ); document.write( "25*36\r
\n" ); document.write( "\n" ); document.write( "30*30\r
\n" ); document.write( "\n" ); document.write( "(-1)*(-900)\r
\n" ); document.write( "\n" ); document.write( "(-2)*(-450)\r
\n" ); document.write( "\n" ); document.write( "(-3)*(-300)\r
\n" ); document.write( "\n" ); document.write( "(-4)*(-225)\r
\n" ); document.write( "\n" ); document.write( "(-5)*(-180)\r
\n" ); document.write( "\n" ); document.write( "(-6)*(-150)\r
\n" ); document.write( "\n" ); document.write( "(-9)*(-100)\r
\n" ); document.write( "\n" ); document.write( "(-10)*(-90)\r
\n" ); document.write( "\n" ); document.write( "(-12)*(-75)\r
\n" ); document.write( "\n" ); document.write( "(-15)*(-60)\r
\n" ); document.write( "\n" ); document.write( "(-18)*(-50)\r
\n" ); document.write( "\n" ); document.write( "(-20)*(-45)\r
\n" ); document.write( "\n" ); document.write( "(-25)*(-36)\r
\n" ); document.write( "\n" ); document.write( "(-30)*(-30)\r
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\n" ); document.write( "\n" ); document.write( "note: remember two negative numbers multiplied together make a positive number\r
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\n" ); document.write( "\n" ); document.write( "Now which of these pairs add to -60? Lets make a table of all of the pairs of factors we multiplied and see which two numbers add to -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 this list we can see that -30 and -30 add up to -60 and multiply to 900\r
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\n" ); document.write( "\n" ); document.write( "Now looking at the expression $\"25b%5E2-60b%2B36\"$ , replace $\"-60b\"$ with $\"-30b%2B-30b\"$ (notice $\"-30b%2B-30b\"$ adds up to $\"-60b\"$ . So it is equivalent to $\"-60b\"$ )\r
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\n" ); document.write( "\n" ); document.write( " $\"25b%5E2%2Bhighlight%28-30b%2B-30b%29%2B36\"$ \r
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\n" ); document.write( "\n" ); document.write( "Now let's factor $\"25b%5E2-30b-30b%2B36\"$ by grouping:\r
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\n" ); document.write( "\n" ); document.write( " $\"%2825b%5E2-30b%29%2B%28-30b%2B36%29\"$ Group like terms\r
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\n" ); document.write( "\n" ); document.write( " $\"5b%285b-6%29-6%285b-6%29\"$ Factor out the GCF of $\"5b\"$ out of the first group. Factor out the GCF of $\"-6\"$ out of the second group\r
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\n" ); document.write( "\n" ); document.write( " $\"%285b-6%29%285b-6%29\"$ Since we have a common term of $\"5b-6\"$ , we can combine like terms\r
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\n" ); document.write( "\n" ); document.write( "So $\"25b%5E2-30b-30b%2B36\"$ factors to $\"%285b-6%29%285b-6%29\"$ \r
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\n" ); document.write( "\n" ); document.write( "So this also means that $\"25b%5E2-60b%2B36\"$ factors to $\"%285b-6%29%285b-6%29\"$ (since $\"25b%5E2-60b%2B36\"$ is equivalent to $\"25b%5E2-30b-30b%2B36\"$ )\r
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\n" ); document.write( "\n" ); document.write( "note: $\"%285b-6%29%285b-6%29\"$ is equivalent to $\"%285b-6%29%5E2\"$ since the term $\"5b-6\"$ occurs twice. So $\"25b%5E2-60b%2B36\"$ also factors to $\"%285b-6%29%5E2\"$ \r
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\n" ); document.write( "\n" ); document.write( "So our expression goes from $\"3%2825b%5E2-60b%2B36%29\"$ and factors further to $\"3%285b-6%29%5E2\"$ \r
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\n" ); document.write( "\n" ); document.write( "So $\"75b%5E2-180b%2B108\"$ factors to $\"3%285b-6%29%5E2\"$ \r
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