document.write( "Question 202022: Could somebody guide me about this which is about the relation between the roots and the coefficients?
\n" ); document.write( "For what values of m will the equation \"x%5E2-4x%2B7=m%28x-1%29\" have
\n" ); document.write( "a) one root the reciprocal of the other,
\n" ); document.write( "b) one root equal zero
\n" ); document.write( "c) equal roots
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Algebra.Com's Answer #152311 by Edwin McCravy(20056)\"\" \"About 
You can put this solution on YOUR website!
Could somebody guide me about this which is about the relation between the roots and the coefficients?\r
\n" ); document.write( "\n" ); document.write( "For what values of m will the equation \"x%5E2-4x%2B7=m%28x-1%29\" have
\n" ); document.write( "a) one root the reciprocal of the other,
\n" ); document.write( "b) one root equal zero
\n" ); document.write( "c) equal roots
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\r\n" );
document.write( "To do any of those we have to first get\r\n" );
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document.write( "\"x%5E2-4x%2B7=m%28x-1%29\" in the form \"x%5E2%2BAx%2BB=0\"\r\n" );
document.write( "\"x%5E2-4x%2B7=mx-m\"\r\n" );
document.write( "\"x%5E2-4x-mx%2B7%2Bm=0\"\r\n" );
document.write( "\"x%5E2-%284%2Bm%29x%2B%287%2Bm%29=0\"\r\n" );
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document.write( "----------------\r\n" );
document.write( "

\n" ); document.write( "a) one root the reciprocal of the other
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\r\n" );
document.write( "Suppose one root is r and the other is \"1%2Fr\", then\r\n" );
document.write( "the quadratic equation with this property and leading\r\n" );
document.write( "coefficient 1 is found this way:\r\n" );
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document.write( "\"%28x-r%29%28x-1%2Fr%29=0\"\r\n" );
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document.write( "\"x%5E2-%28r%2B1%2Fr%29x%2B1=0\"\r\n" );
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document.write( "Therefore this must be the same equation as \r\n" );
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document.write( "\"x%5E2-%284%2Bm%29x%2B%287%2Bm%29=0\"\r\n" );
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document.write( "So we equate like parts:\r\n" );
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document.write( "\"system%28-%28r%2B1%2Fr%29=-%284%2Bm%29%2C1=%287%2Bm%29%29\"\r\n" );
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document.write( "or simplifying:\r\n" );
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document.write( "\"system%28r%2B1%2Fr=4%2Bm%2C-6=m%29\"\r\n" );
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document.write( "To check that, we substitute \"-6\" \r\n" );
document.write( "for \"m\":\r\n" );
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document.write( "\"x%5E2-4x%2B7=m%28x-1%29\"\r\n" );
document.write( "\"x%5E2-4x%2B7=-6%28x-1%29\"\r\n" );
document.write( "\"x%5E2-4x%2B7=-6x%2B6\"\r\n" );
document.write( "\"x%5E2%2B2x%2B1=0\"\r\n" );
document.write( "\"%28x%2B1%29%28x%2B1%29=0\"\r\n" );
document.write( "\"x%2B1=0\", \"x%2B1=0\"    \r\n" );
document.write( "\"x=-1\"  , \"x=-1\"\r\n" );
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document.write( "So both roots are equal and -1, but -1 is the\r\n" );
document.write( "reciprocal of -1.\r\n" );
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document.write( "So the answer to (a) is \"m=-6\"\r\n" );
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document.write( "

\n" ); document.write( "b) one root equal zero
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\r\n" );
document.write( "Suppose one root is 0 and the other is r, then\r\n" );
document.write( "the quadratic equation with this property and leading\r\n" );
document.write( "coefficient 1 is found this way:\r\n" );
document.write( "\r\n" );
document.write( "\"%28x-0%29%28x-r%29=0\"\r\n" );
document.write( "\"x%28x-r%29=0\"\r\n" );
document.write( "\"x%5E2-rx=0\"\r\n" );
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document.write( "Therefore this must be the same equation as \r\n" );
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document.write( "\"x%5E2-%284%2Bm%29x%2B%287%2Bm%29=0\"\r\n" );
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document.write( "So we equate like parts:\r\n" );
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document.write( "\"system%28-r=-%284%2Bm%29%2C0=%287%2Bm%29%29\"\r\n" );
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document.write( "Simplifying:\r\n" );
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document.write( "\"system%28r=4%2Bm%2C-7=m%29\"\r\n" );
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document.write( "So we see that m must be -7.\r\n" );
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document.write( "To check that, we substitute \"-7\" \r\n" );
document.write( "for \"m\":\r\n" );
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document.write( "\"x%5E2-4x%2B7=m%28x-1%29\"\r\n" );
document.write( "\"x%5E2-4x%2B7=-7%28x-1%29\"\r\n" );
document.write( "\"x%5E2-4x%2B7=-7x%2B7\"\r\n" );
document.write( "\"x%5E2%2B3x=0\"\r\n" );
document.write( "\"x%28x%2B3%29=0\"\r\n" );
document.write( "\"x=0\", \"x%2B3=0\"    \r\n" );
document.write( "           \"x=-3\"\r\n" );
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document.write( "One root is 0, so we are correct.\r\n" );
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document.write( "So the answer to (b) is \"m=-7\"\r\n" );
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document.write( "

\n" ); document.write( "c) equal roots
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\r\n" );
document.write( "Suppose one root is r and the other is also r, then\r\n" );
document.write( "the quadratic equation with this property and leading\r\n" );
document.write( "coefficient 1 is found this way:\r\n" );
document.write( "\r\n" );
document.write( "\"%28x-r%29%28x-r%29=0\"\r\n" );
document.write( "\"%28x-r%29%28x-r%29=0\"\r\n" );
document.write( "\"x%5E2-2rx%2Br%5E2=0\"\r\n" );
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document.write( "Therefore this must be the same equation as \r\n" );
document.write( "\r\n" );
document.write( "\"x%5E2-%284%2Bm%29x%2B%287%2Bm%29=0\"\r\n" );
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document.write( "So we equate like parts:\r\n" );
document.write( "\r\n" );
document.write( "\"system%28-2r=-%284%2Bm%29%2Cr%5E2=%287%2Bm%29%29\"\r\n" );
document.write( "\r\n" );
document.write( "Simplifying:\r\n" );
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document.write( "\"system%282r=4%2Bm%2Cr%5E2=7%2Bm%29\"\r\n" );
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document.write( "Solve both equations for m\r\n" );
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document.write( "\"system%28m=2r-4%2Cm=r%5E2-7%29\"\r\n" );
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document.write( "Equate the right sides since both equal m:\r\n" );
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document.write( "\"2r-4=r%5E2-7\"\r\n" );
document.write( "\"-r%5E2%2B2r%2B3=0\"\r\n" );
document.write( "\"r%5E2-2r-3=0\"\r\n" );
document.write( "\"%28r%2B1%29%28r-3%29=0\"\r\n" );
document.write( "\"r%2B1=0\", \"r-3=0\"    \r\n" );
document.write( "\"r=-1\" , \"r=3\"\r\n" );
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document.write( "Substituting \"r=-1\" into\r\n" );
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document.write( "\"m=2r-4\"\r\n" );
document.write( "\"m=2%28-1%29-4\"\r\n" );
document.write( "\"m=-2-4\"\r\n" );
document.write( "\"m=-6\"\r\n" );
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document.write( "Substituting \"r=3\" into\r\n" );
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document.write( "\"m=2r-4\"\r\n" );
document.write( "\"m=2%283%29-4\"\r\n" );
document.write( "\"m=6-4\"\r\n" );
document.write( "\"m=2\"\r\n" );
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document.write( "We don't need to check m=-6 for that was\r\n" );
document.write( "the value of m in part (a), and we knew \r\n" );
document.write( "that in that case the roots were not only \r\n" );
document.write( "reciprocals but they were also equal.\r\n" );
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document.write( "Checking m=2\r\n" );
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document.write( "\"x%5E2-4x%2B7=m%28x-1%29\"\r\n" );
document.write( "\"x%5E2-4x%2B7=2%28x-1%29\"\r\n" );
document.write( "\"x%5E2-4x%2B7=2x-2\"\r\n" );
document.write( "\"x%5E2-6x%2B9=0\"\r\n" );
document.write( "\"%28x-3%29%28x-3%29=0\"\r\n" );
document.write( "\"x-3=0\", \"x-3=0\"    \r\n" );
document.write( "\"x=3\"  , \"x=3\"\r\n" );
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document.write( "So there are two answers to (c),\r\n" );
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document.write( "\"m=-6\" and \"m=2\".\r\n" );
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document.write( "Edwin

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