document.write( "Question 1092372: Show that the roots of the equation (x-p)(x-q)=2 are real and distinct for all real values of p and q. \n" ); document.write( "
Algebra.Com's Answer #706979 by Edwin McCravy(20055)\"\" \"About 
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document.write( "        (x-p)(x-q) = 2\r\n" );
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document.write( " x² - qx - px + pq = 2\r\n" );
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document.write( "  x² - (q+p)x + pq = 2\r\n" );
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document.write( "x² - (q+p)x + pq-2 = 0\r\n" );
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document.write( "We must show that the discriminant is always positive.\r\n" );
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document.write( "Discriminant = b² - 4ac = [-(q+p)]² - 4(1)(pq-2)\r\n" );
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document.write( "We simplify this discriminant\r\n" );
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document.write( "(q+p)² - 4(pq-2)\r\n" );
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document.write( "(q+p)² - 4pq + 8\r\n" );
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document.write( "q² + 2pq + p² - 4pq + 8\r\n" );
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document.write( "q² - 2pq + p² + 8\r\n" );
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document.write( "(q - p)² + 8\r\n" );
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document.write( "This will always be positive because it is greater \r\n" );
document.write( "than or equal to 8.\r\n" );
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document.write( "Therefore the roots are real and distinct.\r\n" );
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document.write( "Edwin

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