document.write( "Question 1196201: 
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document.write( "Suppose x is a positive number such that \"x%5E2=1-x\". There is a unique choice of whole numbers p and q so that \"x%5E8=p-qx\". Find p+q.\r\n" );
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document.write( "My attempt:\r\n" );
document.write( "I know that if \"x%5E2=1-x\", then \"x%5E8=%281-x%29%5E4\", which simplifies to \"1+-+4x+%2B+6x%5E2+-+4x%5E3+%2Bx%5E4\". Then what?\r\n" );
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Algebra.Com's Answer #828940 by ikleyn(52798)\"\" \"About 
You can put this solution on YOUR website!
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\n" ); document.write( "\n" ); document.write( "            It is  VERY  nice problem,  admitting  BEATIFUL  solution.\r
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document.write( "If x^2 = 1-x, then\r\n" );
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document.write( "   x^8 = (1-x)^4 = 1 - 4x + 6x^2 - 4x^3 + x^4.\r\n" );
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document.write( "The idea is to replace in this expression, and in all expressions that follow, x^2 by (1-x) \r\n" );
document.write( "everywhere, where it is possible, and as many times as possible, until you get the desired \r\n" );
document.write( "expression of degree 1 (one).\r\n" );
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document.write( "                        See how it works\r\n" );
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document.write( "    x^8 = 1 - 4x + [6(1-x)] - [4x*(1-x)] + [(1-x)*(1-x)] = I continue =\r\n" );
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document.write( "        = 1 - 4x + 6 - 6x - 4x + 4x^2 + 1 - 2x + x^2 = 8 - 14x + 5x^2 = I replace x^2 by (1-x) again = \r\n" );
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document.write( "        = 8 - 14x + 5*(1-x) = 8 - 16x + 5 - 5x = 13 - 21x.    \r\n" );
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document.write( "    So, p = 13, q = 21  and  p + q = 13 + 21 = 34.      ANSWER\r\n" );
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\n" ); document.write( "\n" ); document.write( "This method is called  \" the lowering of a degree \"  method.\r
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\n" ); document.write( "\n" ); document.write( "It can be solved by different methods,  but this one is a  \" true delight \".\r
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