document.write( "Question 1105564: (x+y)(x^2+y^2)=40
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Algebra.Com's Answer #720419 by Edwin McCravy(20060)\"\" \"About 
You can put this solution on YOUR website!
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document.write( "He's no more help than the back of the book. LOL\r\n" );
document.write( "She didn't give the complex solutions.\r\n" );
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document.write( "\"system%28%28x%2By%29%28x%5E2%2By%5E2%29=40%2C%0D%0A%28x-y%29%28x%5E2-y%5E2%29=16%29\"\r\n" );
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document.write( "Notice that if we interchange x and y, the system of equations\r\n" );
document.write( "simplifies to the same system, so if (x,y)=(p,q) is a solution,\r\n" );
document.write( "then so is (x,y)=(q,p) \r\n" );
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document.write( "Let y = kx\r\n" );
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document.write( "\"system%28x%281%2Bk%29x%5E2%281%2Bk%5E2%29=40%2C%0D%0Ax%281-k%29x%5E2%281-k%5E2%29=16%29\"\r\n" );
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document.write( "\"system%28x%5E3%281%2Bk%29%281%2Bk%5E2%29=40%2C%0D%0Ax%5E3%281-k%29%281-k%5E2%29=16%29\"\r\n" );
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document.write( "Solve each for x³\r\n" );
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document.write( "Equate the expressions for x³\r\n" );
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document.write( "\"40%2F%281%2Bk%29%281%2Bk%5E2%29=16%2F%281-k%29%281-k%5E2%29\"\r\n" );
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document.write( "Divide both sides by 8\r\n" );
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document.write( "\"5%2F%281%2Bk%29%281%2Bk%5E2%29=2%2F%281-k%29%281-k%5E2%29\"\r\n" );
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document.write( "Cross-multiply:\r\n" );
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document.write( "\"2%281%2Bk%29%281%2Bk%5E2%29=5%281-k%29%281-k%5E2%29\"\r\n" );
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document.write( "Factor 1-k² as the difference of squares:\r\n" );
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document.write( "\"2%281%2Bk%29%281%2Bk%5E2%29=5%281-k%29%281-k%29%281%2Bk%29\"\r\n" );
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document.write( "Get 0 on the right\r\n" );
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document.write( "\"2%281%2Bk%29%281%2Bk%5E2%29-5%281-k%29%281-k%29%281%2Bk%29=0\"\r\n" );
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document.write( "Factor out common factor (1+k)\r\n" );
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document.write( "\"%281%2Bk%29%282%281%2Bk%5E2%29%5E%22%22-5%281-k%29%281-k%29%5E%22%22%29=0\"\r\n" );
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document.write( "\"%281%2Bk%29%282%2B2k%5E2-5%281-2k%2Bk%5E2%29%29=0\"\r\n" );
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document.write( "\"%281%2Bk%29%282%2B2k%5E2-5%2B10k-5k%5E2%29=0\"\r\n" );
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document.write( "\"%281%2Bk%29%28-3k%5E2+%2B+10k+-+3%29=0\"\r\n" );
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document.write( "\"%281%2Bk%29%28-3k%5E2+%2B+10k+-+3%29=0\"\r\n" );
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document.write( "Use the zero factor property:\r\n" );
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document.write( "1+k = 0;    -3k²+10k-3 = 0\r\n" );
document.write( "  k = -1;    3k²-10k+3 = 0\r\n" );
document.write( "           (3k-1)(k-3) = 0  \r\n" );
document.write( "         3k-1 = 0;  k-3 = 0\r\n" );
document.write( "           3k = 1;    k = 3\r\n" );
document.write( "            k = 1/3\r\n" );
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document.write( "We substitute those three values for k in:\r\n" );
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document.write( "\"x%5E3%281-k%29%281-k%5E2%29=16\"\r\n" );
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document.write( "Substituting k=-1\r\n" );
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document.write( "\"x%5E3%281-%28-1%29%5E%22%22%29%281-%28-1%29%5E2%29=16\"\r\n" );
document.write( "\"x%5E3%281%2B1%29%281-1%29=16\"\r\n" );
document.write( "\"0=16\", not possible:\r\n" );
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document.write( "Substituting k=1/3\r\n" );
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document.write( "\"x%5E3%281-%281%2F3%29%5E%22%22%29%281-%281%2F3%29%5E2%29=16\"\r\n" );
document.write( "\"x%5E3%282%2F3%29%281-1%2F9%29=16\" \r\n" );
document.write( "\"x%5E3%282%2F3%29%288%2F9%29=16\"\r\n" );
document.write( "\"x%5E3%2816%2F27%29=16\"\r\n" );
document.write( "\"x%5E3%2816%29=16%2A27\"\r\n" );
document.write( "\"x%5E3=27\"\r\n" );
document.write( "\"x%5E3-27=0\"\r\n" );
document.write( "\"%28x-3%29%28x%5E2%2B3x%2B9%29=0+\"}}}\r\n" );
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document.write( "Use zero-factor property:\r\n" );
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document.write( "x=3; x²+3x+9=0\r\n" );
document.write( "     \"x+=+%28-3+%2B-+sqrt%28+3%5E2-4%2A9%2A1+%29%29%2F%282%2A1%29+\"\r\n" );
document.write( "     \"x+=+%28-3+%2B-+sqrt%289-36%29%29%2F2+\"\r\n" );
document.write( "     \"x+=+%28-3+%2B-+sqrt%28-27%29%29%2F2+\"\r\n" );
document.write( "     \"x+=+%28-3+%2B-+i%2Asqrt%2827%29%29%2F2+\"\r\n" );
document.write( "     \"x+=+%28-3+%2B-+i%2Asqrt%289%2A3%29%29%2F2+\"\r\n" );
document.write( "     \"x+=+%28-3+%2B-+3i%2Asqrt%283%29%29%2F2+\"\r\n" );
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document.write( "And since y = kx = (1/3)x\r\n" );
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document.write( "when x = 3, y = (1/3)(1) = 1\r\n" );
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document.write( "So one solution is (x,y) = (3,1), and by swapping x and y, we\r\n" );
document.write( "have\r\n" );
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document.write( " (x,y) = (1,3)\r\n" );
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document.write( "and when \"x+=+%28-3+%2B-+3i%2Asqrt%283%29%29%2F2+\",  \"y+=+%281%2F3%29%28%28-3+%2B-+3i%2Asqrt%283%29%29%2F2%29+=+%28-1+%2B-+i%2Asqrt%283%29%29%2F2+\"\r\n" );
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document.write( "So two more solutions are\r\n" );
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document.write( "and\r\n" );
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document.write( "And two more solutions by swapping x and y are\r\n" );
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document.write( "and\r\n" );
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document.write( "Edwin
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