document.write( "Question 148141: how do i factor 20c(squared+17c-63r squared
\n" ); document.write( " \n" ); document.write( "

Algebra.Com's Answer #108559 by jim_thompson5910(35256) $\"\"$ $\"About$

You can put this solution on YOUR website!
\r
\n" ); document.write( "\n" ); document.write( "Looking at $\"20c%5E2%2B17c-63r%5E2\"$ we can see that the first term is $\"20c%5E2\"$ and the last term is $\"-63r%5E2\"$ where the coefficients are 20 and -63 respectively.\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "Now multiply the first coefficient 20 and the last coefficient -63 to get -1260. Now what two numbers multiply to -1260 and add to the middle coefficient 17? Let's list all of the factors of -1260:\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "Factors of -1260:\r
\n" ); document.write( "\n" ); document.write( "1,2,3,4,5,6,7,9,10,12,14,15,18,20,21,28,30,35,36,42,45,60,63,70,84,90,105,126,140,180,210,252,315,420,630,1260\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "-1,-2,-3,-4,-5,-6,-7,-9,-10,-12,-14,-15,-18,-20,-21,-28,-30,-35,-36,-42,-45,-60,-63,-70,-84,-90,-105,-126,-140,-180,-210,-252,-315,-420,-630,-1260 ...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 -1260\r
\n" ); document.write( "\n" ); document.write( "(1)*(-1260)\r
\n" ); document.write( "\n" ); document.write( "(2)*(-630)\r
\n" ); document.write( "\n" ); document.write( "(3)*(-420)\r
\n" ); document.write( "\n" ); document.write( "(4)*(-315)\r
\n" ); document.write( "\n" ); document.write( "(5)*(-252)\r
\n" ); document.write( "\n" ); document.write( "(6)*(-210)\r
\n" ); document.write( "\n" ); document.write( "(7)*(-180)\r
\n" ); document.write( "\n" ); document.write( "(9)*(-140)\r
\n" ); document.write( "\n" ); document.write( "(10)*(-126)\r
\n" ); document.write( "\n" ); document.write( "(12)*(-105)\r
\n" ); document.write( "\n" ); document.write( "(14)*(-90)\r
\n" ); document.write( "\n" ); document.write( "(15)*(-84)\r
\n" ); document.write( "\n" ); document.write( "(18)*(-70)\r
\n" ); document.write( "\n" ); document.write( "(20)*(-63)\r
\n" ); document.write( "\n" ); document.write( "(21)*(-60)\r
\n" ); document.write( "\n" ); document.write( "(28)*(-45)\r
\n" ); document.write( "\n" ); document.write( "(30)*(-42)\r
\n" ); document.write( "\n" ); document.write( "(35)*(-36)\r
\n" ); document.write( "\n" ); document.write( "(-1)*(1260)\r
\n" ); document.write( "\n" ); document.write( "(-2)*(630)\r
\n" ); document.write( "\n" ); document.write( "(-3)*(420)\r
\n" ); document.write( "\n" ); document.write( "(-4)*(315)\r
\n" ); document.write( "\n" ); document.write( "(-5)*(252)\r
\n" ); document.write( "\n" ); document.write( "(-6)*(210)\r
\n" ); document.write( "\n" ); document.write( "(-7)*(180)\r
\n" ); document.write( "\n" ); document.write( "(-9)*(140)\r
\n" ); document.write( "\n" ); document.write( "(-10)*(126)\r
\n" ); document.write( "\n" ); document.write( "(-12)*(105)\r
\n" ); document.write( "\n" ); document.write( "(-14)*(90)\r
\n" ); document.write( "\n" ); document.write( "(-15)*(84)\r
\n" ); document.write( "\n" ); document.write( "(-18)*(70)\r
\n" ); document.write( "\n" ); document.write( "(-20)*(63)\r
\n" ); document.write( "\n" ); document.write( "(-21)*(60)\r
\n" ); document.write( "\n" ); document.write( "(-28)*(45)\r
\n" ); document.write( "\n" ); document.write( "(-30)*(42)\r
\n" ); document.write( "\n" ); document.write( "(-35)*(36)\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "note: remember, the product of a negative and a positive number is a negative number\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "Now which of these pairs add to 17? Lets make a table of all of the pairs of factors we multiplied and see which two numbers add to 17\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "\n" ); document.write( "\n" ); document.write( "

First Number	Second Number	Sum
1	-1260	1+(-1260)=-1259
2	-630	2+(-630)=-628
3	-420	3+(-420)=-417
4	-315	4+(-315)=-311
5	-252	5+(-252)=-247
6	-210	6+(-210)=-204
7	-180	7+(-180)=-173
9	-140	9+(-140)=-131
10	-126	10+(-126)=-116
12	-105	12+(-105)=-93
14	-90	14+(-90)=-76
15	-84	15+(-84)=-69
18	-70	18+(-70)=-52
20	-63	20+(-63)=-43
21	-60	21+(-60)=-39
28	-45	28+(-45)=-17
30	-42	30+(-42)=-12
35	-36	35+(-36)=-1
-1	1260	-1+1260=1259
-2	630	-2+630=628
-3	420	-3+420=417
-4	315	-4+315=311
-5	252	-5+252=247
-6	210	-6+210=204
-7	180	-7+180=173
-9	140	-9+140=131
-10	126	-10+126=116
-12	105	-12+105=93
-14	90	-14+90=76
-15	84	-15+84=69
-18	70	-18+70=52
-20	63	-20+63=43
-21	60	-21+60=39
-28	45	-28+45=17
-30	42	-30+42=12
-35	36	-35+36=1

\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "From this list we can see that -28 and 45 add up to 17 and multiply to -1260\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "Now looking at the expression $\"20c%5E2%2B17c-63r%5E2\"$ , replace $\"17c\"$ with $\"-28c%2B45c\"$ (notice $\"-28c%2B45c\"$ adds up to $\"17c\"$ . So it is equivalent to $\"17c\"$ )\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( " $\"20c%5E2%2Bhighlight%28-28c%2B45c%29%2B-63r%5E2\"$ \r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "Now let's factor $\"20c%5E2-28c%2B45c-63r%5E2\"$ by grouping:\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( " $\"%2820c%5E2-28c%29%2B%2845c-63r%5E2%29\"$ Group like terms\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( " $\"4c%285c-7%29%2B9%285c-7r%5E2%29\"$ Factor out the GCF of $\"4c\"$ out of the first group. Factor out the GCF of $\"9\"$ out of the second group\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "So this also means that $\"20c%5E2%2B17c-63r%5E2\"$ factors to $\"%284c%2B9%29%285c-7%29\"$ (since $\"20c%5E2%2B17c-63r%5E2\"$ is equivalent to $\"20c%5E2-28c%2B45c-63r%5E2\"$ )\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( " Answer:\r
\n" ); document.write( "\n" ); document.write( "So $\"20c%5E2%2B17c-63r%5E2\"$ factors to $\"%284c%2B9%29%285c-7%29\"$
\n" ); document.write( " \n" ); document.write( "

\n" );