document.write( "Question 1012813: If 1/(a-(-w))-(-1)/(b-(-w))-(-1)/(c -(-w))-(-1)/(d-(-w))=2w^2 and 1/(a-(-w^2))-(-1)/(b-(-w^2))-(-1)/(c-(-w^2))-(-1)/(d-(-w^2))=2w then prove that 1/(a-(-1))-(-1)/(b-(-1))-(-1)/(c-(-1))-(-1)/(d-(-1))=2
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Algebra.Com's Answer #628915 by ikleyn(52788) $\"\"$ $\"About$

You can put this solution on YOUR website!
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document.write( "Now your formulas look much better.\r\n" );
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document.write( "Thank you for following my instructions.\r\n" );
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document.write( "Now tell us please what is  in these formulas.\r\n" );
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document.write( "OK, very good. \r\n" );
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document.write( "So,  w =  =   is this complex number.\r\n" );
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document.write( "Let me rewrite what is given and what you want to prove in an equivalent, but simpler form:\r\n" );
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document.write( "If \r\n" );
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document.write( " =           (1)    and \r\n" );
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document.write( " =     (2)    then prove that \r\n" );
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document.write( " = .\r\n" );
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document.write( "Let us add the formulas (1) and (2) term-by-term. I will start with the pair of addends with the symbol \"a\", and will simplify it step by step:\r\n" );
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document.write( " +  =             (write with the common denominator and then add the numerators. You will get)\r\n" );
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document.write( "=  =  =   (now recall that  = 1. So, replace  by 1 in the last formula. You will get)\r\n" );
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document.write( "=  =   (now use the fact that for your special number \"w\" there is the identity   = -1. \r\n" );
document.write( "                          So, substitute it into the last formula. You will get)\r\n" );
document.write( "= .\r\n" );
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document.write( "OK. It looks like I can not simplify it further.\r\n" );
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document.write( "You can make this trick with every other pair of addends containing the symbols \"b\", \"c\" and \"d\". \r\n" );
document.write( "In this way you will obtain similar formulas.\r\n" );
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document.write( "Now, when you add right sides of these long formulas, you will get .  It  equals  -2  based on what was noted above.\r\n" );
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document.write( "Finally, after adding these long formulas and simplifications, you get\r\n" );
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document.write( " +  +  +  = 2.\r\n" );
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document.write( "It is not exactly what you requested.\r\n" );
document.write( "But it is what our mother-nature allows us to do.\r\n" );
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