document.write( "Question 340411: factor completely\r
\n" ); document.write( "\n" ); document.write( " 150v^2+360vp+216p^2 \n" ); document.write( "

Algebra.Com's Answer #243917 by jim_thompson5910(35256) $\"\"$ $\"About$

You can put this solution on YOUR website!
\r
\n" ); document.write( "\n" ); document.write( " $\"150v%5E2%2B360vp%2B216p%5E2\"$ Start with the given expression\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( " $\"6%2825v%5E2%2B60vp%2B36p%5E2%29\"$ Factor out the GCF $\"6\"$ \r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "Now let's focus on the inner expression $\"25v%5E2%2B60vp%2B36p%5E2\"$ \r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "------------------------------------------------------------\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "Looking at $\"25v%5E2%2B60vp%2B36p%5E2\"$ we can see that the first term is $\"25v%5E2\"$ and the last term is $\"36p%5E2\"$ where the coefficients are 25 and 36 respectively.\r
\n" ); document.write( "
\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
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "
\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
\n" ); document.write( "
\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
\n" ); document.write( "
\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
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "note: remember two negative numbers multiplied together make a positive number\r
\n" ); document.write( "
\n" ); document.write( "
\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
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "\n" ); document.write( "\n" ); document.write( "

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

\r
\n" ); document.write( "
\n" ); document.write( "
\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
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "Now looking at the expression $\"25v%5E2%2B60vp%2B36p%5E2\"$ , replace $\"60vp\"$ with $\"30vp%2B30vp\"$ (notice $\"30vp%2B30vp\"$ adds up to $\"60vp\"$ . So it is equivalent to $\"60vp\"$ )\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( " $\"25v%5E2%2Bhighlight%2830vp%2B30vp%29%2B36p%5E2\"$ \r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "Now let's factor $\"25v%5E2%2B30vp%2B30vp%2B36p%5E2\"$ by grouping:\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( " $\"%2825v%5E2%2B30vp%29%2B%2830vp%2B36p%5E2%29\"$ Group like terms\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( " $\"5v%285v%2B6p%29%2B6p%285v%2B6p%29\"$ Factor out the GCF of $\"5v\"$ out of the first group. Factor out the GCF of $\"6p\"$ out of the second group\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( " $\"%285v%2B6p%29%285v%2B6p%29\"$ Since we have a common term of $\"5v%2B6p\"$ , we can combine like terms\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "So $\"25v%5E2%2B30vp%2B30vp%2B36p%5E2\"$ factors to $\"%285v%2B6p%29%285v%2B6p%29\"$ \r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "So this also means that $\"25v%5E2%2B60vp%2B36p%5E2\"$ factors to $\"%285v%2B6p%29%285v%2B6p%29\"$ (since $\"25v%5E2%2B60vp%2B36p%5E2\"$ is equivalent to $\"25v%5E2%2B30vp%2B30vp%2B36p%5E2\"$ )\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "note: $\"%285v%2B6p%29%285v%2B6p%29\"$ is equivalent to $\"%285v%2B6p%29%5E2\"$ since the term $\"5v%2B6p\"$ occurs twice. So $\"25v%5E2%2B60vp%2B36p%5E2\"$ also factors to $\"%285v%2B6p%29%5E2\"$ \r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "------------------------------------------------------------\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "So our expression goes from $\"6%2825v%5E2%2B60vp%2B36p%5E2%29\"$ and factors further to $\"6%285v%2B6p%29%5E2\"$ \r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "------------------\r
\n" ); document.write( "\n" ); document.write( "Answer:\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "So $\"150v%5E2%2B360vp%2B216p%5E2\"$ factors to $\"6%285v%2B6p%29%5E2\"$ \r
\n" ); document.write( "\n" ); document.write( " \r
\n" ); document.write( "\n" ); document.write( "If you need more help, email me at jim_thompson5910@hotmail.com\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "Also, feel free to check out my tutoring website\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "Jim \n" ); document.write( "

\n" );