document.write( "Question 303580: Factor:
\n" ); document.write( "42-13r+r^2
\n" ); document.write( "I don't even know where to begin with this problem!
\n" ); document.write( "I am also lost when it comes to finding the variation constant and an equation of variation where y varies directly as x and y=4 when x=1. I know that y=kx, but everytime I do the math I come out with y=4x...am I on the right track?? Thank you so much for your time!! \n" ); document.write( "

Algebra.Com's Answer #217552 by jim_thompson5910(35256) $\"\"$ $\"About$

You can put this solution on YOUR website!
# 1\r
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\n" ); document.write( "\n" ); document.write( " $\"42-13r%2Br%5E2\"$ Start with the given expression.\r
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\n" ); document.write( "\n" ); document.write( " $\"r%5E2-13r%2B42\"$ Rearrange the terms\r
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\n" ); document.write( "\n" ); document.write( "Looking at the expression $\"r%5E2-13r%2B42\"$ , we can see that the first coefficient is $\"1\"$ , the second coefficient is $\"-13\"$ , and the last term is $\"42\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Now multiply the first coefficient $\"1\"$ by the last term $\"42\"$ to get $\"%281%29%2842%29=42\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Now the question is: what two whole numbers multiply to $\"42\"$ (the previous product) and add to the second coefficient $\"-13\"$ ?\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 $\"42\"$ (the previous product).\r
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\n" ); document.write( "\n" ); document.write( "Factors of $\"42\"$ :\r
\n" ); document.write( "\n" ); document.write( "1,2,3,6,7,14,21,42\r
\n" ); document.write( "\n" ); document.write( "-1,-2,-3,-6,-7,-14,-21,-42\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 $\"42\"$ .\r
\n" ); document.write( "\n" ); document.write( "1*42 = 42
\n" ); document.write( "2*21 = 42
\n" ); document.write( "3*14 = 42
\n" ); document.write( "6*7 = 42
\n" ); document.write( "(-1)*(-42) = 42
\n" ); document.write( "(-2)*(-21) = 42
\n" ); document.write( "(-3)*(-14) = 42
\n" ); document.write( "(-6)*(-7) = 42\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 $\"-13\"$ :\r
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First Number	Second Number	Sum
1	42	1+42=43
2	21	2+21=23
3	14	3+14=17
6	7	6+7=13
-1	-42	-1+(-42)=-43
-2	-21	-2+(-21)=-23
-3	-14	-3+(-14)=-17
-6	-7	-6+(-7)=-13

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\n" ); document.write( "\n" ); document.write( "From the table, we can see that the two numbers $\"-6\"$ and $\"-7\"$ add to $\"-13\"$ (the middle coefficient).\r
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\n" ); document.write( "\n" ); document.write( "So the two numbers $\"-6\"$ and $\"-7\"$ both multiply to $\"42\"$ and add to $\"-13\"$ \r
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\n" ); document.write( "\n" ); document.write( "Now replace the middle term $\"-13r\"$ with $\"-6r-7r\"$ . Remember, $\"-6\"$ and $\"-7\"$ add to $\"-13\"$ . So this shows us that $\"-6r-7r=-13r\"$ .\r
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\n" ); document.write( "\n" ); document.write( " $\"r%5E2%2Bhighlight%28-6r-7r%29%2B42\"$ Replace the second term $\"-13r\"$ with $\"-6r-7r\"$ .\r
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\n" ); document.write( "\n" ); document.write( " $\"%28r%5E2-6r%29%2B%28-7r%2B42%29\"$ Group the terms into two pairs.\r
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\n" ); document.write( "\n" ); document.write( " $\"r%28r-6%29%2B%28-7r%2B42%29\"$ Factor out the GCF $\"r\"$ from the first group.\r
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\n" ); document.write( "\n" ); document.write( " $\"r%28r-6%29-7%28r-6%29\"$ Factor out $\"7\"$ 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( " $\"%28r-7%29%28r-6%29\"$ Combine like terms. Or factor out the common term $\"r-6\"$ \r
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\n" ); document.write( "\n" ); document.write( "Answer:\r
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\n" ); document.write( "\n" ); document.write( "So $\"42-13r%2Br%5E2\"$ factors to $\"%28r-7%29%28r-6%29\"$ .\r
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\n" ); document.write( "\n" ); document.write( "In other words, $\"42-13r%2Br%5E2=%28r-7%29%28r-6%29\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Note: you can check the answer by expanding $\"%28r-7%29%28r-6%29\"$ to get $\"42-13r%2Br%5E2\"$ or by graphing the original expression and the answer (the two graphs should be identical).\r
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\n" ); document.write( "\n" ); document.write( "# 2 You have the correct equation $\"y=4x\"$ . Notice how when x=1, $\"y=4%281%29=4\"$ or simply $\"y=4\"$
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