document.write( "Question 640704: 35x^2+3xy-54y^2 \n" ); document.write( "

Algebra.Com's Answer #403372 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 $\"35x%5E2%2B3xy-54y%5E2\"$ , we can see that the first coefficient is $\"35\"$ , the second coefficient is $\"3\"$ , and the last coefficient is $\"-54\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Now multiply the first coefficient $\"35\"$ by the last coefficient $\"-54\"$ to get $\"%2835%29%28-54%29=-1890\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Now the question is: what two whole numbers multiply to $\"-1890\"$ (the previous product) and add to the second coefficient $\"3\"$ ?\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 $\"-1890\"$ (the previous product).\r
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\n" ); document.write( "\n" ); document.write( "Factors of $\"-1890\"$ :\r
\n" ); document.write( "\n" ); document.write( "1,2,3,5,6,7,9,10,14,15,18,21,27,30,35,42,45,54,63,70,90,105,126,135,189,210,270,315,378,630,945,1890\r
\n" ); document.write( "\n" ); document.write( "-1,-2,-3,-5,-6,-7,-9,-10,-14,-15,-18,-21,-27,-30,-35,-42,-45,-54,-63,-70,-90,-105,-126,-135,-189,-210,-270,-315,-378,-630,-945,-1890\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 $\"-1890\"$ .\r
\n" ); document.write( "\n" ); document.write( "1*(-1890) = -1890
\n" ); document.write( "2*(-945) = -1890
\n" ); document.write( "3*(-630) = -1890
\n" ); document.write( "5*(-378) = -1890
\n" ); document.write( "6*(-315) = -1890
\n" ); document.write( "7*(-270) = -1890
\n" ); document.write( "9*(-210) = -1890
\n" ); document.write( "10*(-189) = -1890
\n" ); document.write( "14*(-135) = -1890
\n" ); document.write( "15*(-126) = -1890
\n" ); document.write( "18*(-105) = -1890
\n" ); document.write( "21*(-90) = -1890
\n" ); document.write( "27*(-70) = -1890
\n" ); document.write( "30*(-63) = -1890
\n" ); document.write( "35*(-54) = -1890
\n" ); document.write( "42*(-45) = -1890
\n" ); document.write( "(-1)*(1890) = -1890
\n" ); document.write( "(-2)*(945) = -1890
\n" ); document.write( "(-3)*(630) = -1890
\n" ); document.write( "(-5)*(378) = -1890
\n" ); document.write( "(-6)*(315) = -1890
\n" ); document.write( "(-7)*(270) = -1890
\n" ); document.write( "(-9)*(210) = -1890
\n" ); document.write( "(-10)*(189) = -1890
\n" ); document.write( "(-14)*(135) = -1890
\n" ); document.write( "(-15)*(126) = -1890
\n" ); document.write( "(-18)*(105) = -1890
\n" ); document.write( "(-21)*(90) = -1890
\n" ); document.write( "(-27)*(70) = -1890
\n" ); document.write( "(-30)*(63) = -1890
\n" ); document.write( "(-35)*(54) = -1890
\n" ); document.write( "(-42)*(45) = -1890\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 $\"3\"$ :\r
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First Number	Second Number	Sum
1	-1890	1+(-1890)=-1889
2	-945	2+(-945)=-943
3	-630	3+(-630)=-627
5	-378	5+(-378)=-373
6	-315	6+(-315)=-309
7	-270	7+(-270)=-263
9	-210	9+(-210)=-201
10	-189	10+(-189)=-179
14	-135	14+(-135)=-121
15	-126	15+(-126)=-111
18	-105	18+(-105)=-87
21	-90	21+(-90)=-69
27	-70	27+(-70)=-43
30	-63	30+(-63)=-33
35	-54	35+(-54)=-19
42	-45	42+(-45)=-3
-1	1890	-1+1890=1889
-2	945	-2+945=943
-3	630	-3+630=627
-5	378	-5+378=373
-6	315	-6+315=309
-7	270	-7+270=263
-9	210	-9+210=201
-10	189	-10+189=179
-14	135	-14+135=121
-15	126	-15+126=111
-18	105	-18+105=87
-21	90	-21+90=69
-27	70	-27+70=43
-30	63	-30+63=33
-35	54	-35+54=19
-42	45	-42+45=3

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\n" ); document.write( "\n" ); document.write( "From the table, we can see that the two numbers $\"-42\"$ and $\"45\"$ add to $\"3\"$ (the middle coefficient).\r
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\n" ); document.write( "\n" ); document.write( "So the two numbers $\"-42\"$ and $\"45\"$ both multiply to $\"-1890\"$ and add to $\"3\"$ \r
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\n" ); document.write( "\n" ); document.write( "Now replace the middle term $\"3xy\"$ with $\"-42xy%2B45xy\"$ . Remember, $\"-42\"$ and $\"45\"$ add to $\"3\"$ . So this shows us that $\"-42xy%2B45xy=3xy\"$ .\r
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\n" ); document.write( "\n" ); document.write( " $\"35x%5E2%2Bhighlight%28-42xy%2B45xy%29-54y%5E2\"$ Replace the second term $\"3xy\"$ with $\"-42xy%2B45xy\"$ .\r
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\n" ); document.write( "\n" ); document.write( " $\"%2835x%5E2-42xy%29%2B%2845xy-54y%5E2%29\"$ Group the terms into two pairs.\r
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\n" ); document.write( "\n" ); document.write( " $\"7x%285x-6y%29%2B%2845xy-54y%5E2%29\"$ Factor out the GCF $\"7x\"$ from the first group.\r
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\n" ); document.write( "\n" ); document.write( " $\"7x%285x-6y%29%2B9y%285x-6y%29\"$ Factor out $\"9y\"$ 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( " $\"%287x%2B9y%29%285x-6y%29\"$ Combine like terms. Or factor out the common term $\"5x-6y\"$ \r
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\n" ); document.write( "\n" ); document.write( "Answer:\r
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\n" ); document.write( "\n" ); document.write( "So $\"35x%5E2%2B3xy-54y%5E2\"$ factors to $\"%287x%2B9y%29%285x-6y%29\"$ .\r
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\n" ); document.write( "\n" ); document.write( "In other words, $\"35x%5E2%2B3xy-54y%5E2=%287x%2B9y%29%285x-6y%29\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Note: you can check the answer by expanding $\"%287x%2B9y%29%285x-6y%29\"$ to get $\"35x%5E2%2B3xy-54y%5E2\"$ or by graphing the original expression and the answer (the two graphs should be identical).
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