document.write( "Question 112031: I was having alot of trouble with this problem (factor completely)\r
\n" ); document.write( "\n" ); document.write( "6h^2+17hk+10k^2\r
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Algebra.Com's Answer #81717 by jim_thompson5910(35256) $\"\"$ $\"About$

You can put this solution on YOUR website!
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\n" ); document.write( "\n" ); document.write( "Looking at $\"6h%5E2%2B17hk%2B10k%5E2\"$ we can see that the first term is $\"6h%5E2\"$ and the last term is $\"10k%5E2\"$ where the coefficients are 6 and 10 respectively.\r
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\n" ); document.write( "\n" ); document.write( "Now multiply the first coefficient 6 and the last coefficient 10 to get 60. Now what two numbers multiply to 60 and add to the middle coefficient 17? Let's list all of the factors of 60:\r
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\n" ); document.write( "\n" ); document.write( "Factors of 60:\r
\n" ); document.write( "\n" ); document.write( "1,2,3,4,5,6,10,12,15,20,30,60\r
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\n" ); document.write( "\n" ); document.write( "-1,-2,-3,-4,-5,-6,-10,-12,-15,-20,-30,-60 ...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 60\r
\n" ); document.write( "\n" ); document.write( "1*60\r
\n" ); document.write( "\n" ); document.write( "2*30\r
\n" ); document.write( "\n" ); document.write( "3*20\r
\n" ); document.write( "\n" ); document.write( "4*15\r
\n" ); document.write( "\n" ); document.write( "5*12\r
\n" ); document.write( "\n" ); document.write( "6*10\r
\n" ); document.write( "\n" ); document.write( "(-1)*(-60)\r
\n" ); document.write( "\n" ); document.write( "(-2)*(-30)\r
\n" ); document.write( "\n" ); document.write( "(-3)*(-20)\r
\n" ); document.write( "\n" ); document.write( "(-4)*(-15)\r
\n" ); document.write( "\n" ); document.write( "(-5)*(-12)\r
\n" ); document.write( "\n" ); document.write( "(-6)*(-10)\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 17? Lets make a table of all of the pairs of factors we multiplied and see which two numbers add to 17\r
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First Number	Second Number	Sum
1	60	1+60=61
2	30	2+30=32
3	20	3+20=23
4	15	4+15=19
5	12	5+12=17
6	10	6+10=16
-1	-60	-1+(-60)=-61
-2	-30	-2+(-30)=-32
-3	-20	-3+(-20)=-23
-4	-15	-4+(-15)=-19
-5	-12	-5+(-12)=-17
-6	-10	-6+(-10)=-16

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\n" ); document.write( "\n" ); document.write( "From this list we can see that 5 and 12 add up to 17 and multiply to 60\r
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\n" ); document.write( "\n" ); document.write( "Now looking at the expression $\"6h%5E2%2B17hk%2B10k%5E2\"$ , replace $\"17hk\"$ with $\"5hk%2B12hk\"$ (notice $\"5hk%2B12hk\"$ adds up to $\"17hk\"$ . So it is equivalent to $\"17hk\"$ )\r
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\n" ); document.write( "\n" ); document.write( " $\"6h%5E2%2Bhighlight%285hk%2B12hk%29%2B10k%5E2\"$ \r
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\n" ); document.write( "\n" ); document.write( "Now let's factor $\"6h%5E2%2B5hk%2B12hk%2B10k%5E2\"$ by grouping:\r
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\n" ); document.write( "\n" ); document.write( " $\"%286h%5E2%2B5hk%29%2B%2812hk%2B10k%5E2%29\"$ Group like terms\r
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\n" ); document.write( "\n" ); document.write( " $\"h%286h%2B5k%29%2B2k%286h%2B5k%29\"$ Factor out the GCF of $\"h\"$ out of the first group. Factor out the GCF of $\"2k\"$ out of the second group\r
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\n" ); document.write( "\n" ); document.write( " $\"%28h%2B2k%29%286h%2B5k%29\"$ Since we have a common term of $\"6h%2B5k\"$ , we can combine like terms\r
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\n" ); document.write( "\n" ); document.write( "So $\"6h%5E2%2B5hk%2B12hk%2B10k%5E2\"$ factors to $\"%28h%2B2k%29%286h%2B5k%29\"$ \r
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\n" ); document.write( "\n" ); document.write( "So this also means that $\"6h%5E2%2B17hk%2B10k%5E2\"$ factors to $\"%28h%2B2k%29%286h%2B5k%29\"$ (since $\"6h%5E2%2B17hk%2B10k%5E2\"$ is equivalent to $\"6h%5E2%2B5hk%2B12hk%2B10k%5E2\"$ )
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