document.write( "Question 187009: please factor and solve
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\n" ); document.write( "12z^2+z=6 \n" ); document.write( "

Algebra.Com's Answer #140170 by jim_thompson5910(35256) $\"\"$ $\"About$

You can put this solution on YOUR website!
$\"12z%5E2%2Bz=6\"$ Start with the given equation.\r
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\n" ); document.write( "\n" ); document.write( " $\"12z%5E2%2Bz-6=0\"$ Subtract 6 from both sides.\r
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\n" ); document.write( "\n" ); document.write( "Let's factor the left side:\r
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\n" ); document.write( "\n" ); document.write( "Looking at the expression $\"12z%5E2%2Bz-6\"$ , we can see that the first coefficient is $\"12\"$ , the second coefficient is $\"1\"$ , and the last term is $\"-6\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Now multiply the first coefficient $\"12\"$ by the last term $\"-6\"$ to get $\"%2812%29%28-6%29=-72\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Now the question is: what two whole numbers multiply to $\"-72\"$ (the previous product) and add to the second coefficient $\"1\"$ ?\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 $\"-72\"$ (the previous product).\r
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\n" ); document.write( "\n" ); document.write( "Factors of $\"-72\"$ :\r
\n" ); document.write( "\n" ); document.write( "1,2,3,4,6,8,9,12,18,24,36,72\r
\n" ); document.write( "\n" ); document.write( "-1,-2,-3,-4,-6,-8,-9,-12,-18,-24,-36,-72\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 $\"-72\"$ .\r
\n" ); document.write( "\n" ); document.write( "1*(-72)
\n" ); document.write( "2*(-36)
\n" ); document.write( "3*(-24)
\n" ); document.write( "4*(-18)
\n" ); document.write( "6*(-12)
\n" ); document.write( "8*(-9)
\n" ); document.write( "(-1)*(72)
\n" ); document.write( "(-2)*(36)
\n" ); document.write( "(-3)*(24)
\n" ); document.write( "(-4)*(18)
\n" ); document.write( "(-6)*(12)
\n" ); document.write( "(-8)*(9)\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 $\"1\"$ :\r
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First Number	Second Number	Sum
1	-72	1+(-72)=-71
2	-36	2+(-36)=-34
3	-24	3+(-24)=-21
4	-18	4+(-18)=-14
6	-12	6+(-12)=-6
8	-9	8+(-9)=-1
-1	72	-1+72=71
-2	36	-2+36=34
-3	24	-3+24=21
-4	18	-4+18=14
-6	12	-6+12=6
-8	9	-8+9=1

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\n" ); document.write( "\n" ); document.write( "From the table, we can see that the two numbers $\"-8\"$ and $\"9\"$ add to $\"1\"$ (the middle coefficient).\r
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\n" ); document.write( "\n" ); document.write( "So the two numbers $\"-8\"$ and $\"9\"$ both multiply to $\"-72\"$ and add to $\"1\"$ \r
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\n" ); document.write( "\n" ); document.write( "Now replace the middle term $\"1z\"$ with $\"-8z%2B9z\"$ . Remember, $\"-8\"$ and $\"9\"$ add to $\"1\"$ . So this shows us that $\"-8z%2B9z=1z\"$ .\r
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\n" ); document.write( "\n" ); document.write( " $\"12z%5E2%2Bhighlight%28-8z%2B9z%29-6\"$ Replace the second term $\"1z\"$ with $\"-8z%2B9z\"$ .\r
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\n" ); document.write( "\n" ); document.write( " $\"%2812z%5E2-8z%29%2B%289z-6%29\"$ Group the terms into two pairs.\r
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\n" ); document.write( "\n" ); document.write( " $\"4z%283z-2%29%2B%289z-6%29\"$ Factor out the GCF $\"4z\"$ from the first group.\r
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\n" ); document.write( "\n" ); document.write( " $\"4z%283z-2%29%2B3%283z-2%29\"$ Factor out $\"3\"$ 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( " $\"%284z%2B3%29%283z-2%29\"$ Combine like terms. Or factor out the common term $\"3z-2\"$ \r
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\n" ); document.write( "\n" ); document.write( "So $\"12z%5E2%2Bz-6\"$ factors to $\"%284z%2B3%29%283z-2%29\"$ .\r
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\n" ); document.write( "\n" ); document.write( "So $\"12z%5E2%2Bz-6=0\"$ becomes $\"%284z%2B3%29%283z-2%29=0\"$ \r
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\n" ); document.write( "\n" ); document.write( "Now set each factor equal to zero:\r
\n" ); document.write( "\n" ); document.write( " $\"4z%2B3=0\"$ or $\"3z-2=0\"$ \r
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\n" ); document.write( "\n" ); document.write( " $\"z=-3%2F4\"$ or $\"z=2%2F3\"$ Now solve for z in each case\r
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\n" ); document.write( "\n" ); document.write( "So our answers are \r
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\n" ); document.write( "\n" ); document.write( " $\"z=-3%2F4\"$ or $\"z=2%2F3\"$ \n" ); document.write( "

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