document.write( "Question 507702: 6. Completely factor the following expression:
\n" ); document.write( "36a^2 – 84ab + 49b^2\r
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\n" ); document.write( "\n" ); document.write( "7. Completely factor the following expression:
\n" ); document.write( "9y^2 – 16z^2 \r
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\n" ); document.write( "\n" ); document.write( "I do not understand please help me!I am desperate!! \n" ); document.write( "

Algebra.Com's Answer #340693 by jim_thompson5910(35256) $\"\"$ $\"About$

You can put this solution on YOUR website!
# 6\r
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\n" ); document.write( "\n" ); document.write( "Looking at the expression $\"36a%5E2-84ab%2B49b%5E2\"$ , 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 $\"-84ab\"$ with $\"-42ab-42ab\"$ . Remember, $\"-42\"$ and $\"-42\"$ add to $\"-84\"$ . So this shows us that $\"-42ab-42ab=-84ab\"$ .\r
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\n" ); document.write( "\n" ); document.write( " $\"36a%5E2%2Bhighlight%28-42ab-42ab%29%2B49b%5E2\"$ Replace the second term $\"-84ab\"$ with $\"-42ab-42ab\"$ .\r
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\n" ); document.write( "\n" ); document.write( " $\"%2836a%5E2-42ab%29%2B%28-42ab%2B49b%5E2%29\"$ Group the terms into two pairs.\r
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\n" ); document.write( "\n" ); document.write( " $\"6a%286a-7b%29%2B%28-42ab%2B49b%5E2%29\"$ Factor out the GCF $\"6a\"$ from the first group.\r
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\n" ); document.write( "\n" ); document.write( " $\"6a%286a-7b%29-7b%286a-7b%29\"$ Factor out $\"-7b\"$ from the second group.\r
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\n" ); document.write( "\n" ); document.write( " $\"%286a-7b%29%286a-7b%29\"$ Factor out the GCF $\"6a-7b\"$ \r
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\n" ); document.write( "\n" ); document.write( " $\"%286a-7b%29%5E2\"$ Condense the terms.\r
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\n" ); document.write( "\n" ); document.write( "So $\"36a%5E2-84ab%2B49b%5E2\"$ completely factors to $\"%286a-7b%29%5E2\"$ \r
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\n" ); document.write( "\n" ); document.write( "In other words, $\"36a%5E2-84ab%2B49b%5E2=%286a-7b%29%5E2\"$ \r
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\n" ); document.write( "\n" ); document.write( "# 7\r
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\n" ); document.write( "\n" ); document.write( " $\"9y%5E2-16z%5E2\"$ Start with the given expression.\r
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\n" ); document.write( "\n" ); document.write( " $\"%283y%29%5E2-16z%5E2\"$ Rewrite $\"9y%5E2\"$ as $\"%283y%29%5E2\"$ .\r
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\n" ); document.write( "\n" ); document.write( " $\"%283y%29%5E2-%284z%29%5E2\"$ Rewrite $\"16z%5E2\"$ as $\"%284z%29%5E2\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Notice how we have a difference of squares $\"A%5E2-B%5E2\"$ where in this case $\"A=3y\"$ and $\"B=4z\"$ .\r
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\n" ); document.write( "\n" ); document.write( "So let's use the difference of squares formula $\"A%5E2-B%5E2=%28A%2BB%29%28A-B%29\"$ to factor the expression:\r
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\n" ); document.write( "\n" ); document.write( " $\"A%5E2-B%5E2=%28A%2BB%29%28A-B%29\"$ Start with the difference of squares formula.\r
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\n" ); document.write( "\n" ); document.write( " $\"%283y%29%5E2-%284z%29%5E2=%283y%2B4z%29%283y-4z%29\"$ Plug in $\"A=3y\"$ and $\"B=4z\"$ .\r
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\n" ); document.write( "\n" ); document.write( "So this shows us that $\"9y%5E2-16z%5E2\"$ factors to $\"%283y%2B4z%29%283y-4z%29\"$ .\r
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\n" ); document.write( "\n" ); document.write( "In other words $\"9y%5E2-16z%5E2=%283y%2B4z%29%283y-4z%29\"$ .\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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