document.write( "Question 183211: \"find all integers m for which y^2+my+50 can be factored?\" \n" ); document.write( "

Algebra.Com's Answer #137556 by jim_thompson5910(35256) $\"\"$ $\"About$

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First, multiply the first coefficient 1 and the last term 50 to get 50. Now list the factors of 50\r
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\n" ); document.write( "\n" ); document.write( "Factors of 50:\r
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\n" ); document.write( "\n" ); document.write( "-1,-2,-5,-10,-25,-50\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 $\"50\"$ .\r
\n" ); document.write( "\n" ); document.write( "1*50
\n" ); document.write( "2*25
\n" ); document.write( "5*10
\n" ); document.write( "(-1)*(-50)
\n" ); document.write( "(-2)*(-25)
\n" ); document.write( "(-5)*(-10)\r
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\n" ); document.write( "\n" ); document.write( "Now add up each paired factor (ie 1+50=51, 2+25=27, etc..):\r
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First Number	Second Number	Sum
1	50	1+50=51
2	25	2+25=27
5	10	5+10=15
-1	-50	-1+(-50)=-51
-2	-25	-2+(-25)=-27
-5	-10	-5+(-10)=-15

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\n" ); document.write( "\n" ); document.write( "All of the numbers in the last column are possible values for the value of \"m\". Remember, you can only factor $\"ax%5E2%2Bbx%2Bc\"$ only if the factors of $\"ac\"$ add to \"b\"\r
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\n" ); document.write( "\n" ); document.write( "So the possible values for \"m\" are: 51, 27, 15, -51, -27, -15\r
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\n" ); document.write( "\n" ); document.write( "Note: all of these values of \"m\" are straight from that third column.\r
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\n" ); document.write( "\n" ); document.write( "This means that the following quadratics are factorable:\r
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\n" ); document.write( "\n" ); document.write( " $\"y%5E2%2B51y%2B50\"$ , \r
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\n" ); document.write( "\n" ); document.write( " $\"y%5E2%2B27y%2B50\"$ ,\r
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\n" ); document.write( "\n" ); document.write( " $\"y%5E2%2B15y%2B50\"$ ,\r
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\n" ); document.write( "\n" ); document.write( " $\"y%5E2-51y%2B50\"$ ,\r
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\n" ); document.write( "\n" ); document.write( " $\"y%5E2-27y%2B50\"$ , and \r
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\n" ); document.write( "\n" ); document.write( " $\"y%5E2-15y%2B50\"$
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