document.write( "Question 506441: Please Help Me solve This problem: factor 252w^2+588wf+343f^2 \n" ); document.write( "

Algebra.Com's Answer #340159 by jim_thompson5910(35256) $\"\"$ $\"About$

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\n" ); document.write( "\n" ); document.write( " $\"252w%5E2%2B588wf%2B343f%5E2\"$ Start with the given expression.\r
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\n" ); document.write( "\n" ); document.write( " $\"7%2836w%5E2%2B84fw%2B49f%5E2%29\"$ Factor out the GCF $\"7\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Now let's try to factor the inner expression $\"36w%5E2%2B84fw%2B49f%5E2\"$ \r
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\n" ); document.write( "\n" ); document.write( "Looking at the expression $\"36w%5E2%2B84fw%2B49f%5E2\"$ , we can see that the first coefficient is $\"36\"$ , the second coefficient is $\"84\"$ , and the last coefficient is $\"49\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Now multiply the first coefficient $\"36\"$ by the last coefficient $\"49\"$ to get $\"%2836%29%2849%29=1764\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Now the question is: what two whole numbers multiply to $\"1764\"$ (the previous product) and add to the second coefficient $\"84\"$ ?\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 $\"1764\"$ (the previous product).\r
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\n" ); document.write( "\n" ); document.write( "Factors of $\"1764\"$ :\r
\n" ); document.write( "\n" ); document.write( "1,2,3,4,6,7,9,12,14,18,21,28,36,42,49,63,84,98,126,147,196,252,294,441,588,882,1764\r
\n" ); document.write( "\n" ); document.write( "-1,-2,-3,-4,-6,-7,-9,-12,-14,-18,-21,-28,-36,-42,-49,-63,-84,-98,-126,-147,-196,-252,-294,-441,-588,-882,-1764\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 $\"1764\"$ .\r
\n" ); document.write( "\n" ); document.write( "1*1764 = 1764
\n" ); document.write( "2*882 = 1764
\n" ); document.write( "3*588 = 1764
\n" ); document.write( "4*441 = 1764
\n" ); document.write( "6*294 = 1764
\n" ); document.write( "7*252 = 1764
\n" ); document.write( "9*196 = 1764
\n" ); document.write( "12*147 = 1764
\n" ); document.write( "14*126 = 1764
\n" ); document.write( "18*98 = 1764
\n" ); document.write( "21*84 = 1764
\n" ); document.write( "28*63 = 1764
\n" ); document.write( "36*49 = 1764
\n" ); document.write( "42*42 = 1764
\n" ); document.write( "(-1)*(-1764) = 1764
\n" ); document.write( "(-2)*(-882) = 1764
\n" ); document.write( "(-3)*(-588) = 1764
\n" ); document.write( "(-4)*(-441) = 1764
\n" ); document.write( "(-6)*(-294) = 1764
\n" ); document.write( "(-7)*(-252) = 1764
\n" ); document.write( "(-9)*(-196) = 1764
\n" ); document.write( "(-12)*(-147) = 1764
\n" ); document.write( "(-14)*(-126) = 1764
\n" ); document.write( "(-18)*(-98) = 1764
\n" ); document.write( "(-21)*(-84) = 1764
\n" ); document.write( "(-28)*(-63) = 1764
\n" ); document.write( "(-36)*(-49) = 1764
\n" ); document.write( "(-42)*(-42) = 1764\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 $\"84\"$ :\r
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First Number	Second Number	Sum
1	1764	1+1764=1765
2	882	2+882=884
3	588	3+588=591
4	441	4+441=445
6	294	6+294=300
7	252	7+252=259
9	196	9+196=205
12	147	12+147=159
14	126	14+126=140
18	98	18+98=116
21	84	21+84=105
28	63	28+63=91
36	49	36+49=85
42	42	42+42=84
-1	-1764	-1+(-1764)=-1765
-2	-882	-2+(-882)=-884
-3	-588	-3+(-588)=-591
-4	-441	-4+(-441)=-445
-6	-294	-6+(-294)=-300
-7	-252	-7+(-252)=-259
-9	-196	-9+(-196)=-205
-12	-147	-12+(-147)=-159
-14	-126	-14+(-126)=-140
-18	-98	-18+(-98)=-116
-21	-84	-21+(-84)=-105
-28	-63	-28+(-63)=-91
-36	-49	-36+(-49)=-85
-42	-42	-42+(-42)=-84

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\n" ); document.write( "\n" ); document.write( "From the table, we can see that the two numbers $\"42\"$ and $\"42\"$ add to $\"84\"$ (the middle coefficient).\r
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\n" ); document.write( "\n" ); document.write( "So the two numbers $\"42\"$ and $\"42\"$ both multiply to $\"1764\"$ and add to $\"84\"$ \r
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\n" ); document.write( "\n" ); document.write( "Now replace the middle term $\"84fw\"$ with $\"42fw%2B42fw\"$ . Remember, $\"42\"$ and $\"42\"$ add to $\"84\"$ . So this shows us that $\"42fw%2B42fw=84fw\"$ .\r
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\n" ); document.write( "\n" ); document.write( " $\"36w%5E2%2Bhighlight%2842fw%2B42fw%29%2B49f%5E2\"$ Replace the second term $\"84fw\"$ with $\"42fw%2B42fw\"$ .\r
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\n" ); document.write( "\n" ); document.write( " $\"%2836w%5E2%2B42fw%29%2B%2842fw%2B49f%5E2%29\"$ Group the terms into two pairs.\r
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\n" ); document.write( "\n" ); document.write( " $\"6w%286w%2B7f%29%2B%2842fw%2B49f%5E2%29\"$ Factor out the GCF $\"6w\"$ from the first group.\r
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\n" ); document.write( "\n" ); document.write( " $\"6w%286w%2B7f%29%2B7f%286w%2B7f%29\"$ Factor out $\"7f\"$ 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( " $\"%286w%2B7f%29%286w%2B7f%29\"$ Combine like terms. Or factor out the common term $\"6w%2B7f\"$ \r
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\n" ); document.write( "\n" ); document.write( " $\"%286w%2B7f%29%5E2\"$ Condense the terms.\r
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\n" ); document.write( "\n" ); document.write( "So $\"7%2836w%5E2%2B84fw%2B49f%5E2%29\"$ then factors further to $\"7%286w%2B7f%29%5E2\"$ \r
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\n" ); document.write( "\n" ); document.write( "Answer:\r
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\n" ); document.write( "\n" ); document.write( "So $\"252w%5E2%2B588wf%2B343f%5E2\"$ completely factors to $\"7%286w%2B7f%29%5E2\"$ .\r
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\n" ); document.write( "\n" ); document.write( "In other words, $\"252w%5E2%2B588wf%2B343f%5E2=7%286w%2B7f%29%5E2\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Note: you can check the answer by expanding $\"7%286w%2B7f%29%5E2\"$ to get $\"252w%5E2%2B588wf%2B343f%5E2\"$ 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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