document.write( "Question 585384: factor:
\n" ); document.write( "m^2-24m+144 \n" ); document.write( "

Algebra.Com's Answer #373316 by jim_thompson5910(35256) $\"\"$ $\"About$

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\n" ); document.write( "\n" ); document.write( "Looking at the expression $\"m%5E2-24m%2B144\"$ , we can see that the first coefficient is $\"1\"$ , the second coefficient is $\"-24\"$ , and the last term is $\"144\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Now multiply the first coefficient $\"1\"$ by the last term $\"144\"$ to get $\"%281%29%28144%29=144\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Now the question is: what two whole numbers multiply to $\"144\"$ (the previous product) and add to the second coefficient $\"-24\"$ ?\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 $\"144\"$ (the previous product).\r
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\n" ); document.write( "\n" ); document.write( "Factors of $\"144\"$ :\r
\n" ); document.write( "\n" ); document.write( "1,2,3,4,6,8,9,12,16,18,24,36,48,72,144\r
\n" ); document.write( "\n" ); document.write( "-1,-2,-3,-4,-6,-8,-9,-12,-16,-18,-24,-36,-48,-72,-144\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 $\"144\"$ .\r
\n" ); document.write( "\n" ); document.write( "1*144 = 144
\n" ); document.write( "2*72 = 144
\n" ); document.write( "3*48 = 144
\n" ); document.write( "4*36 = 144
\n" ); document.write( "6*24 = 144
\n" ); document.write( "8*18 = 144
\n" ); document.write( "9*16 = 144
\n" ); document.write( "12*12 = 144
\n" ); document.write( "(-1)*(-144) = 144
\n" ); document.write( "(-2)*(-72) = 144
\n" ); document.write( "(-3)*(-48) = 144
\n" ); document.write( "(-4)*(-36) = 144
\n" ); document.write( "(-6)*(-24) = 144
\n" ); document.write( "(-8)*(-18) = 144
\n" ); document.write( "(-9)*(-16) = 144
\n" ); document.write( "(-12)*(-12) = 144\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 $\"-24\"$ :\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 the table, we can see that the two numbers $\"-12\"$ and $\"-12\"$ add to $\"-24\"$ (the middle coefficient).\r
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\n" ); document.write( "\n" ); document.write( "So the two numbers $\"-12\"$ and $\"-12\"$ both multiply to $\"144\"$ and add to $\"-24\"$ \r
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\n" ); document.write( "\n" ); document.write( "Now replace the middle term $\"-24m\"$ with $\"-12m-12m\"$ . Remember, $\"-12\"$ and $\"-12\"$ add to $\"-24\"$ . So this shows us that $\"-12m-12m=-24m\"$ .\r
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\n" ); document.write( "\n" ); document.write( " $\"m%5E2%2Bhighlight%28-12m-12m%29%2B144\"$ Replace the second term $\"-24m\"$ with $\"-12m-12m\"$ .\r
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\n" ); document.write( "\n" ); document.write( " $\"%28m%5E2-12m%29%2B%28-12m%2B144%29\"$ Group the terms into two pairs.\r
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\n" ); document.write( "\n" ); document.write( " $\"m%28m-12%29%2B%28-12m%2B144%29\"$ Factor out the GCF $\"m\"$ from the first group.\r
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\n" ); document.write( "\n" ); document.write( " $\"m%28m-12%29-12%28m-12%29\"$ Factor out $\"12\"$ 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( " $\"%28m-12%29%28m-12%29\"$ Combine like terms. Or factor out the common term $\"m-12\"$ \r
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\n" ); document.write( "\n" ); document.write( " $\"%28m-12%29%5E2\"$ 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 $\"m%5E2-24m%2B144\"$ factors to $\"%28m-12%29%5E2\"$ .\r
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\n" ); document.write( "\n" ); document.write( "In other words, $\"m%5E2-24m%2B144=%28m-12%29%5E2\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Note: you can check the answer by expanding $\"%28m-12%29%5E2\"$ to get $\"m%5E2-24m%2B144\"$ or by graphing the original expression and the answer (the two graphs should be identical).\r
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