document.write( "Question 131907: Factor $\"9s%5E2%2B16-24s\"$ \n" ); document.write( "

Algebra.Com's Answer #96304 by jim_thompson5910(35256) $\"\"$ $\"About$

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$\"9s%5E2%2B16-24s\"$ Start with the given expression\r
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\n" ); document.write( "\n" ); document.write( " $\"9s%5E2-24s%2B16\"$ Rearrange the terms\r
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\n" ); document.write( "\n" ); document.write( "Looking at $\"9s%5E2-24s%2B16\"$ we can see that the first term is $\"9s%5E2\"$ and the last term is $\"16\"$ where the coefficients are 9 and 16 respectively.\r
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\n" ); document.write( "\n" ); document.write( "Now multiply the first coefficient 9 and the last coefficient 16 to get 144. Now what two numbers multiply to 144 and add to the middle coefficient -24? Let's list all of the factors of 144:\r
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\n" ); document.write( "\n" ); document.write( "Factors of 144:\r
\n" ); document.write( "\n" ); document.write( "1,2,3,4,6,8,9,12,16,18,24,36,48,72\r
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\n" ); document.write( "\n" ); document.write( "-1,-2,-3,-4,-6,-8,-9,-12,-16,-18,-24,-36,-48,-72 ...List the negative factors as well. 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 144\r
\n" ); document.write( "\n" ); document.write( "1*144\r
\n" ); document.write( "\n" ); document.write( "2*72\r
\n" ); document.write( "\n" ); document.write( "3*48\r
\n" ); document.write( "\n" ); document.write( "4*36\r
\n" ); document.write( "\n" ); document.write( "6*24\r
\n" ); document.write( "\n" ); document.write( "8*18\r
\n" ); document.write( "\n" ); document.write( "9*16\r
\n" ); document.write( "\n" ); document.write( "12*12\r
\n" ); document.write( "\n" ); document.write( "(-1)*(-144)\r
\n" ); document.write( "\n" ); document.write( "(-2)*(-72)\r
\n" ); document.write( "\n" ); document.write( "(-3)*(-48)\r
\n" ); document.write( "\n" ); document.write( "(-4)*(-36)\r
\n" ); document.write( "\n" ); document.write( "(-6)*(-24)\r
\n" ); document.write( "\n" ); document.write( "(-8)*(-18)\r
\n" ); document.write( "\n" ); document.write( "(-9)*(-16)\r
\n" ); document.write( "\n" ); document.write( "(-12)*(-12)\r
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\n" ); document.write( "\n" ); document.write( "note: remember two negative numbers multiplied together make a positive number\r
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\n" ); document.write( "\n" ); document.write( "Now which of these pairs add to -24? Lets make a table of all of the pairs of factors we multiplied and see which two numbers add to -24\r
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First Number	Second Number	Sum
1	144	1+144=145
2	72	2+72=74
3	48	3+48=51
4	36	4+36=40
6	24	6+24=30
8	18	8+18=26
9	16	9+16=25
12	12	12+12=24
-1	-144	-1+(-144)=-145
-2	-72	-2+(-72)=-74
-3	-48	-3+(-48)=-51
-4	-36	-4+(-36)=-40
-6	-24	-6+(-24)=-30
-8	-18	-8+(-18)=-26
-9	-16	-9+(-16)=-25
-12	-12	-12+(-12)=-24

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\n" ); document.write( "\n" ); document.write( "From this list we can see that -12 and -12 add up to -24 and multiply to 144\r
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\n" ); document.write( "\n" ); document.write( "Now looking at the expression $\"9s%5E2-24s%2B16\"$ , replace $\"-24s\"$ with $\"-12s%2B-12s\"$ (notice $\"-12s%2B-12s\"$ adds up to $\"-24s\"$ . So it is equivalent to $\"-24s\"$ )\r
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\n" ); document.write( "\n" ); document.write( " $\"9s%5E2%2Bhighlight%28-12s%2B-12s%29%2B16\"$ \r
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\n" ); document.write( "\n" ); document.write( "Now let's factor $\"9s%5E2-12s-12s%2B16\"$ by grouping:\r
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\n" ); document.write( "\n" ); document.write( " $\"%289s%5E2-12s%29%2B%28-12s%2B16%29\"$ Group like terms\r
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\n" ); document.write( "\n" ); document.write( " $\"3s%283s-4%29-4%283s-4%29\"$ Factor out the GCF of $\"3s\"$ out of the first group. Factor out the GCF of $\"-4\"$ out of the second group\r
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\n" ); document.write( "\n" ); document.write( " $\"%283s-4%29%283s-4%29\"$ Since we have a common term of $\"3s-4\"$ , we can combine like terms\r
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\n" ); document.write( "\n" ); document.write( "So $\"9s%5E2-12s-12s%2B16\"$ factors to $\"%283s-4%29%283s-4%29\"$ \r
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\n" ); document.write( "\n" ); document.write( "So this also means that $\"9s%5E2-24s%2B16\"$ factors to $\"%283s-4%29%283s-4%29\"$ (since $\"9s%5E2-24s%2B16\"$ is equivalent to $\"9s%5E2-12s-12s%2B16\"$ )\r
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\n" ); document.write( "\n" ); document.write( "note: $\"%283s-4%29%283s-4%29\"$ is equivalent to $\"%283s-4%29%5E2\"$ since the term $\"3s-4\"$ occurs twice. So $\"9s%5E2-24s%2B16\"$ also factors to $\"%283s-4%29%5E2\"$ \r
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\n" ); document.write( "\n" ); document.write( " Answer:\r
\n" ); document.write( "\n" ); document.write( "So $\"9s%5E2-24s%2B16\"$ factors to $\"%283s-4%29%5E2\"$
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