document.write( "Question 638994: 9x^2+48xz+64z^2 \n" ); document.write( "

Algebra.Com's Answer #402505 by jim_thompson5910(35256) $\"\"$ $\"About$

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\n" ); document.write( "\n" ); document.write( "Looking at the expression $\"9x%5E2%2B48xz%2B64z%5E2\"$ , we can see that the first coefficient is $\"9\"$ , the second coefficient is $\"48\"$ , and the last coefficient is $\"64\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Now multiply the first coefficient $\"9\"$ by the last coefficient $\"64\"$ to get $\"%289%29%2864%29=576\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Now the question is: what two whole numbers multiply to $\"576\"$ (the previous product) and add to the second coefficient $\"48\"$ ?\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 $\"576\"$ (the previous product).\r
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\n" ); document.write( "\n" ); document.write( "Factors of $\"576\"$ :\r
\n" ); document.write( "\n" ); document.write( "1,2,3,4,6,8,9,12,16,18,24,32,36,48,64,72,96,144,192,288,576\r
\n" ); document.write( "\n" ); document.write( "-1,-2,-3,-4,-6,-8,-9,-12,-16,-18,-24,-32,-36,-48,-64,-72,-96,-144,-192,-288,-576\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 $\"576\"$ .\r
\n" ); document.write( "\n" ); document.write( "1*576 = 576
\n" ); document.write( "2*288 = 576
\n" ); document.write( "3*192 = 576
\n" ); document.write( "4*144 = 576
\n" ); document.write( "6*96 = 576
\n" ); document.write( "8*72 = 576
\n" ); document.write( "9*64 = 576
\n" ); document.write( "12*48 = 576
\n" ); document.write( "16*36 = 576
\n" ); document.write( "18*32 = 576
\n" ); document.write( "24*24 = 576
\n" ); document.write( "(-1)*(-576) = 576
\n" ); document.write( "(-2)*(-288) = 576
\n" ); document.write( "(-3)*(-192) = 576
\n" ); document.write( "(-4)*(-144) = 576
\n" ); document.write( "(-6)*(-96) = 576
\n" ); document.write( "(-8)*(-72) = 576
\n" ); document.write( "(-9)*(-64) = 576
\n" ); document.write( "(-12)*(-48) = 576
\n" ); document.write( "(-16)*(-36) = 576
\n" ); document.write( "(-18)*(-32) = 576
\n" ); document.write( "(-24)*(-24) = 576\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 $\"48\"$ :\r
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First Number	Second Number	Sum
1	576	1+576=577
2	288	2+288=290
3	192	3+192=195
4	144	4+144=148
6	96	6+96=102
8	72	8+72=80
9	64	9+64=73
12	48	12+48=60
16	36	16+36=52
18	32	18+32=50
24	24	24+24=48
-1	-576	-1+(-576)=-577
-2	-288	-2+(-288)=-290
-3	-192	-3+(-192)=-195
-4	-144	-4+(-144)=-148
-6	-96	-6+(-96)=-102
-8	-72	-8+(-72)=-80
-9	-64	-9+(-64)=-73
-12	-48	-12+(-48)=-60
-16	-36	-16+(-36)=-52
-18	-32	-18+(-32)=-50
-24	-24	-24+(-24)=-48

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\n" ); document.write( "\n" ); document.write( "From the table, we can see that the two numbers $\"24\"$ and $\"24\"$ add to $\"48\"$ (the middle coefficient).\r
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\n" ); document.write( "\n" ); document.write( "So the two numbers $\"24\"$ and $\"24\"$ both multiply to $\"576\"$ and add to $\"48\"$ \r
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\n" ); document.write( "\n" ); document.write( "Now replace the middle term $\"48xz\"$ with $\"24xz%2B24xz\"$ . Remember, $\"24\"$ and $\"24\"$ add to $\"48\"$ . So this shows us that $\"24xz%2B24xz=48xz\"$ .\r
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\n" ); document.write( "\n" ); document.write( " $\"9x%5E2%2Bhighlight%2824xz%2B24xz%29%2B64z%5E2\"$ Replace the second term $\"48xz\"$ with $\"24xz%2B24xz\"$ .\r
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\n" ); document.write( "\n" ); document.write( " $\"%289x%5E2%2B24xz%29%2B%2824xz%2B64z%5E2%29\"$ Group the terms into two pairs.\r
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\n" ); document.write( "\n" ); document.write( " $\"3x%283x%2B8z%29%2B%2824xz%2B64z%5E2%29\"$ Factor out the GCF $\"3x\"$ from the first group.\r
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\n" ); document.write( "\n" ); document.write( " $\"3x%283x%2B8z%29%2B8z%283x%2B8z%29\"$ Factor out $\"8z\"$ 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( " $\"%283x%2B8z%29%283x%2B8z%29\"$ Combine like terms. Or factor out the common term $\"3x%2B8z\"$ \r
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\n" ); document.write( "\n" ); document.write( " $\"%283x%2B8z%29%5E2\"$ Condense the terms.\r
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\n" ); document.write( "\n" ); document.write( "Answer:\r
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\n" ); document.write( "\n" ); document.write( "So $\"9x%5E2%2B48xz%2B64z%5E2\"$ factors to $\"%283x%2B8z%29%5E2\"$ .\r
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\n" ); document.write( "\n" ); document.write( "In other words, $\"9x%5E2%2B48xz%2B64z%5E2=%283x%2B8z%29%5E2\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Note: you can check the answer by expanding $\"%283x%2B8z%29%5E2\"$ to get $\"9x%5E2%2B48xz%2B64z%5E2\"$ or by graphing the original expression and the answer (the two graphs should be identical).
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