document.write( "Question 1132708: If the roots are ax^2+bx+c=0, differ by 1, show that they are (a-b)/2a and -(a+b)/2a, and prove that b^2=a(a+4c) \n" ); document.write( "
Algebra.Com's Answer #749812 by Edwin McCravy(20054)\"\" \"About 
You can put this solution on YOUR website!
If the roots are ax^2+bx+c=0, differ by 1,
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document.write( "Let the roots be r and r+1\r\n" );
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document.write( "\"ar%5E2%2Bbr%2Bc=0\" and \"a%28r%2B1%29%5E2%2Bb%28r%2B1%29%2Bc=0\"\r\n" );
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document.write( "So \"ar%5E2%2Bbr%2Bc=a%28r%2B1%29%5E2%2Bb%28r%2B1%29%2Bc\"\r\n" );
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document.write( "\"ar%5E2%2Bbr%2Bc=a%28r%5E2%2B2r%2B1%29%5E2%2Bbr%2Bb%2Bc\"\r\n" );
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document.write( "\"ar%5E2%2Bbr%2Bc=ar%5E2%2B2ar%2Ba%2Bbr%2Bb%2Bc\"\r\n" );
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document.write( "\"0=2ar%2Ba%2Bb\"\r\n" );
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document.write( "\"2ar%2Ba%2Bb=0\"\r\n" );
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document.write( "\"2ar=+-a-b\"\r\n" );
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document.write( "\"r+=+%28-a-b%29%2F%282a%29\"   <-- that's one root\r\n" );
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document.write( "The other root is r+1, so we add 1 to both sides:\r\n" );
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document.write( "\"r%2B1+=+%28-a-b%29%2F%282a%29%2B1\"\r\n" );
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document.write( "\"r%2B1+=+%28-a-b%29%2F%282a%29%2B%282a%29%2F%282a%29\"\r\n" );
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document.write( "\"r%2B1+=+%28-a-b%2B2a%29%2F%282a%29\"\r\n" );
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document.write( "\"r%2B1+=+%28a-b%29%2F%282a%29\"  <-- that's the other root\r\n" );
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\n" ); document.write( "and prove that b^2=a(a+4c)
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document.write( "Substitute either root in the original.  I'll use\r\n" );
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document.write( "\"%28a-b%29%2F%282a%29\", since it has only one - sign. \r\n" );
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document.write( "\"a%28%28a-b%29%2F%282a%29%29%5E2%2Bb%28%28a-b%29%2F%282a%29%29%2Bc=0\"\r\n" );
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document.write( "\"a%28%28a%5E2-2ab%2Bb%5E2%29%2F%284a%5E2%29%29%2B%28%28ab-b%5E2%29%2F%282a%29%29%2Bc=0\" \r\n" );
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document.write( "\"%28%28a%5E2-2ab%2Bb%5E2%29%2F%284a%29%29%2B%28%28ab-b%5E2%29%2F%282a%29%29%2Bc=0\"\r\n" );
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document.write( "Multiply through by 4a\r\n" );
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document.write( "\"%28a%5E2-2ab%2Bb%5E2%29%2B%282ab-2b%5E2%29%2B4ac=0\"\r\n" );
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document.write( "\"a%5E2-2ab%2Bb%5E2%2B2ab-2b%5E2%2B4ac=0\"\r\n" );
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document.write( "\"a%5E2-b%5E2%2B4ac=0\"\r\n" );
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document.write( "\"a%5E2%2B4ac=b%5E2\"\r\n" );
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document.write( "\"a%28a%2B4c%29=b%5E2\"\r\n" );
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document.write( "\"b%5E2=a%28a%2B4c%29\"\r\n" );
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document.write( "Edwin
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