document.write( "Question 472964: Please solve this quadratic equation by completing the square and factoring:\r
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Algebra.Com's Answer #324290 by jim_thompson5910(35256) $\"\"$ $\"About$

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Completing the square:\r
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\n" ); document.write( "\n" ); document.write( " $\"2m%5E2+%2B+5m+=+12\"$ Start with the given equation.\r
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\n" ); document.write( "\n" ); document.write( " $\"2m%5E2+%2B+5m+-+12=0\"$ Subtract 12 from both sides.\r
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\n" ); document.write( "\n" ); document.write( "Now let's complete the square for the left side.\r
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\n" ); document.write( "\n" ); document.write( " $\"2m%5E2%2B5m-12\"$ Start with the given expression.\r
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\n" ); document.write( "\n" ); document.write( " $\"2%28m%5E2%2B%285%2F2%29m-6%29\"$ Factor out the $\"m%5E2\"$ coefficient $\"2\"$ . This step is very important: the $\"m%5E2\"$ coefficient must be equal to 1.\r
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\n" ); document.write( "\n" ); document.write( "Take half of the $\"m\"$ coefficient $\"5%2F2\"$ to get $\"5%2F4\"$ . In other words, $\"%281%2F2%29%285%2F2%29=5%2F4\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Now square $\"5%2F4\"$ to get $\"25%2F16\"$ . In other words, $\"%285%2F4%29%5E2=%285%2F4%29%285%2F4%29=25%2F16\"$ \r
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\n" ); document.write( "\n" ); document.write( " $\"2%28m%5E2%2B%285%2F2%29m%2Bhighlight%2825%2F16-25%2F16%29-6%29\"$ Now add and subtract $\"25%2F16\"$ inside the parenthesis. Make sure to place this after the \"m\" term. Notice how $\"25%2F16-25%2F16=0\"$ . So the expression is not changed.\r
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\n" ); document.write( "\n" ); document.write( " $\"2%28%28m%5E2%2B%285%2F2%29m%2B25%2F16%29-25%2F16-6%29\"$ Group the first three terms.\r
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\n" ); document.write( "\n" ); document.write( " $\"2%28%28m%2B5%2F4%29%5E2-25%2F16-6%29\"$ Factor $\"m%5E2%2B%285%2F2%29m%2B25%2F16\"$ to get $\"%28m%2B5%2F4%29%5E2\"$ .\r
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\n" ); document.write( "\n" ); document.write( " $\"2%28%28m%2B5%2F4%29%5E2-121%2F16%29\"$ Combine like terms.\r
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\n" ); document.write( "\n" ); document.write( " $\"2%28m%2B5%2F4%29%5E2%2B2%28-121%2F16%29\"$ Distribute.\r
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\n" ); document.write( "\n" ); document.write( " $\"2%28m%2B5%2F4%29%5E2-121%2F8\"$ Multiply.\r
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\n" ); document.write( "\n" ); document.write( "So after completing the square, $\"2m%5E2%2B5m-12\"$ transforms to $\"2%28m%2B5%2F4%29%5E2-121%2F8\"$ . So $\"2m%5E2%2B5m-12=2%28m%2B5%2F4%29%5E2-121%2F8\"$ .\r
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\n" ); document.write( "\n" ); document.write( "So $\"2m%5E2%2B5m-12=0\"$ is equivalent to $\"2%28m%2B5%2F4%29%5E2-121%2F8=0\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Now let's solve $\"2%28m%2B5%2F4%29%5E2-121%2F8=0\"$ \r
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\n" ); document.write( "\n" ); document.write( " $\"2%28m%2B5%2F4%29%5E2-121%2F8=0\"$ Start with the given equation.\r
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\n" ); document.write( "\n" ); document.write( " $\"2%28m%2B5%2F4%29%5E2=0%2B121%2F8\"$ Add $\"121%2F8\"$ to both sides.\r
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\n" ); document.write( "\n" ); document.write( " $\"2%28m%2B5%2F4%29%5E2=121%2F8\"$ Combine like terms.\r
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\n" ); document.write( "\n" ); document.write( " $\"%28m%2B5%2F4%29%5E2=%28121%2F8%29%2F%282%29\"$ Divide both sides by $\"2\"$ .\r
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\n" ); document.write( "\n" ); document.write( " $\"%28m%2B5%2F4%29%5E2=121%2F16\"$ Reduce.\r
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\n" ); document.write( "\n" ); document.write( " $\"m%2B5%2F4=%22%22%2B-sqrt%28121%2F16%29\"$ Take the square root of both sides.\r
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\n" ); document.write( "\n" ); document.write( " $\"m%2B5%2F4=sqrt%28121%2F16%29\"$ or $\"m%2B5%2F4=-sqrt%28121%2F16%29\"$ Break up the \"plus/minus\" to form two equations.\r
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\n" ); document.write( "\n" ); document.write( " $\"m%2B5%2F4=11%2F4\"$ or $\"m%2B5%2F4=-11%2F4\"$ Take the square root of $\"121%2F16\"$ to get $\"11%2F4\"$ .\r
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\n" ); document.write( "\n" ); document.write( " $\"m=-5%2F4%2B11%2F4\"$ or $\"m=-5%2F4-11%2F4\"$ Subtract $\"5%2F4\"$ from both sides.\r
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\n" ); document.write( "\n" ); document.write( " $\"m=3%2F2\"$ or $\"m=-4\"$ Combine like 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 the solutions are $\"m=3%2F2\"$ or $\"m=-4\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Factoring:\r
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\n" ); document.write( "\n" ); document.write( " $\"2m%5E2+%2B+5m+=+12\"$ Start with the given equation.\r
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\n" ); document.write( "\n" ); document.write( " $\"2m%5E2+%2B+5m+-+12=0\"$ Subtract 12 from both sides.\r
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\n" ); document.write( "\n" ); document.write( "Now let's factor:\r
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\n" ); document.write( "\n" ); document.write( "Looking at the expression $\"2m%5E2%2B5m-12\"$ , we can see that the first coefficient is $\"2\"$ , the second coefficient is $\"5\"$ , and the last term is $\"-12\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Now multiply the first coefficient $\"2\"$ by the last term $\"-12\"$ to get $\"%282%29%28-12%29=-24\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Now the question is: what two whole numbers multiply to $\"-24\"$ (the previous product) and add to the second coefficient $\"5\"$ ?\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 $\"-24\"$ (the previous product).\r
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\n" ); document.write( "\n" ); document.write( "Factors of $\"-24\"$ :\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 $\"-24\"$ .\r
\n" ); document.write( "\n" ); document.write( "1*(-24) = -24
\n" ); document.write( "2*(-12) = -24
\n" ); document.write( "3*(-8) = -24
\n" ); document.write( "4*(-6) = -24
\n" ); document.write( "(-1)*(24) = -24
\n" ); document.write( "(-2)*(12) = -24
\n" ); document.write( "(-3)*(8) = -24
\n" ); document.write( "(-4)*(6) = -24\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 $\"5\"$ :\r
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First Number	Second Number	Sum
1	-24	1+(-24)=-23
2	-12	2+(-12)=-10
3	-8	3+(-8)=-5
4	-6	4+(-6)=-2
-1	24	-1+24=23
-2	12	-2+12=10
-3	8	-3+8=5
-4	6	-4+6=2

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\n" ); document.write( "\n" ); document.write( "From the table, we can see that the two numbers $\"-3\"$ and $\"8\"$ add to $\"5\"$ (the middle coefficient).\r
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\n" ); document.write( "\n" ); document.write( "So the two numbers $\"-3\"$ and $\"8\"$ both multiply to $\"-24\"$ and add to $\"5\"$ \r
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\n" ); document.write( "\n" ); document.write( "Now replace the middle term $\"5m\"$ with $\"-3m%2B8m\"$ . Remember, $\"-3\"$ and $\"8\"$ add to $\"5\"$ . So this shows us that $\"-3m%2B8m=5m\"$ .\r
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\n" ); document.write( "\n" ); document.write( " $\"2m%5E2%2Bhighlight%28-3m%2B8m%29-12\"$ Replace the second term $\"5m\"$ with $\"-3m%2B8m\"$ .\r
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\n" ); document.write( "\n" ); document.write( " $\"%282m%5E2-3m%29%2B%288m-12%29\"$ Group the terms into two pairs.\r
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\n" ); document.write( "\n" ); document.write( " $\"m%282m-3%29%2B%288m-12%29\"$ Factor out the GCF $\"m\"$ from the first group.\r
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\n" ); document.write( "\n" ); document.write( " $\"m%282m-3%29%2B4%282m-3%29\"$ Factor out $\"4\"$ 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( " $\"%28m%2B4%29%282m-3%29\"$ Combine like terms. Or factor out the common term $\"2m-3\"$ \r
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\n" ); document.write( "\n" ); document.write( "So $\"2m%5E2%2B5m-12\"$ factors to $\"%28m%2B4%29%282m-3%29\"$ .\r
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\n" ); document.write( "\n" ); document.write( "In other words, $\"2m%5E2%2B5m-12=%28m%2B4%29%282m-3%29\"$ .\r
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\n" ); document.write( "\n" ); document.write( "So $\"2m%5E2%2B5m-12=0\"$ turns into $\"%28m%2B4%29%282m-3%29=0\"$ \r
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\n" ); document.write( "\n" ); document.write( " $\"%28m%2B4%29%282m-3%29=0\"$ Start with the given equation\r
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\n" ); document.write( "\n" ); document.write( " $\"m%2B4=0\"$ or $\"2m-3=0\"$ Use the zero product property\r
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\n" ); document.write( "\n" ); document.write( " $\"m=-4\"$ or $\"m=3%2F2\"$ Solve for 'm' in each equation.\r
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\n" ); document.write( "\n" ); document.write( "So the solutions are $\"m=-4\"$ or $\"m=3%2F2\"$ (which are the same as the ones above)
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