document.write( "Question 156386: show that the equations y=kx-1 and y=x^2+(k-1)x+k^2 have no real solution for all values of k. \n" ); document.write( "
Algebra.Com's Answer #115576 by Edwin McCravy(20054)\"\" \"About 
You can put this solution on YOUR website!
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document.write( "show that the equations \"y=kx-1\" and \"y=x%5E2%2B%28k-1%29x%2Bk%5E2\"\r\n" );
document.write( "have no real common solution for all values of k.\r\n" );
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document.write( "\"system%28y=kx-1%2C+y=x%5E2%2B%28k-1%29x%2Bk%5E2%29\"\r\n" );
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document.write( "Set the right side of the first equation equal to\r\n" );
document.write( "the right side of the second equation, since both\r\n" );
document.write( "equal to y:\r\n" );
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document.write( "\"kx-1=x%5E2%2B%28k-1%29x%2Bk%5E2\"\r\n" );
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document.write( "Distribute to remove the parentheses:\r\n" );
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document.write( "\"kx-1=x%5E2%2Bkx-x%2Bk%5E2\"\r\n" );
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document.write( "Get 0 on the left side by adding \"-kx%2B1\"\r\n" );
document.write( "to both sides:\r\n" );
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document.write( "\"kx-1-kx%2B1=x%5E2%2Bkx-x%2Bk%5E2-kx%2B1\"\r\n" );
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document.write( "\"0=x%5E2-x%2Bk%5E2%2B1\"\r\n" );
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document.write( "Swap sides:\r\n" );
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document.write( "\"x%5E2-x%2Bk%5E2%2B1=0\"\r\n" );
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document.write( "Group the last two terms on the left in \r\n" );
document.write( "parentheses:\r\n" );
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document.write( "We need to find the DISCRIMINANT\r\n" );
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document.write( "\"DISCRIMINANT_OF_ax%5E2%2Bbx%2Bc+=+b%5E2-4ac\"\r\n" );
document.write( "\"x%5E2-x%2B%28k%5E2%2B1%29=0\"\r\n" );
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document.write( "is the same as\r\n" );
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document.write( "\"1x%5E2-1x%2B%28k%5E2%2B1%29=0\"\r\n" );
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document.write( "So \"a=1\", \"b=-1\", \"c=k%5E2%2B1\"\r\n" );
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document.write( "\"DISCRIMINANT=b%5E2-4ac\"\r\n" );
document.write( "\"DISCRIMINANT=%28-1%29%5E2-4%281%29%28k%5E2%2B1%29\"\r\n" );
document.write( "\"DISCRIMINANT=1-4%28k%5E2%2B1%29\"\r\n" );
document.write( "\"DISCRIMINANT=1-4k%5E2-4\"\r\n" );
document.write( "\"DISCRIMINANT=-3-4k%5E2\"\r\n" );
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document.write( "There are no real solutions when the\r\n" );
document.write( "\"DISCRIMINANT%3C0\"\r\n" );
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document.write( "and \"-3-4k%5E2\" is ALWAYS negative,\r\n" );
document.write( "since \"k%5E2\" is never negative.\r\n" );
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document.write( "So there can be no real common solutions \r\n" );
document.write( "for any real value of \"k\".\r\n" );
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document.write( "Edwin

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