document.write( "Question 1208228: If a + b = 1 and a⁴ + b⁴ = 17.
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Algebra.Com's Answer #846518 by ikleyn(52787) $\"\"$ $\"About$

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document.write( "Two solutions can be guessed mentally and instantly:\r\n" );
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document.write( "    (a,b) = (2,-1)  and  (a,b) = (-1,2).    (1)\r\n" );
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document.write( "It can be proved mentally and visually.\r\n" );
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document.write( "The curve x^4 + y^4 = 17  is  similar to the circle  x^2 + y^2 = 17.\r\n" );
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document.write( "It is not a circle, of course, but is a similar convex shape.\r\n" );
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document.write( "The line x + y = 1 is the straight line that cuts this curve x^4 + y^4 = 17.\r\n" );
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document.write( "It is clear, that there are two intersection points, and can not be more than 2 intersection points.\r\n" );
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document.write( "Thus, the listed solutions (1) is the full set of real solutions.\r\n" );
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document.write( "If you want to get an algebra solution, then express\r\n" );
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document.write( "    a = 1 - b\r\n" );
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document.write( "from the first equation and substitute into the second equation.\r\n" );
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document.write( "You will get a polynomial equation  P(b) = 17  for  \"b\"  of degree 4.\r\n" );
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document.write( "Two roots are b = -1 and b = 2.\r\n" );
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document.write( "Hence, two linear divisors are (b+1) and (b-2).\r\n" );
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document.write( "Divide the polynomial  P(b)-17  by the product (b+1)*(b-2).\r\n" );
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document.write( "You will get the quotient as a quadratic polynomial, which has no real roots.\r\n" );
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document.write( "From it, you will conclude that the listed solutions (1) are the only solutions, and there are no other solutions.\r\n" );
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