Algebra.Com's Answer #801485 by ikleyn(52781)![\"\"](\"/images/stars/stars-13.gif\")  
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document.write( "Find an ordered triple (x,y,z) of real numbers satisfying x<= y<= z and the system of equations
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document.write( "sqrt{x} + sqrt{y} + sqrt{z} = 10,
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document.write( "x + y + z = 38,
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document.write( "sqrt{xy} + sqrt{xz} + sqrt{yz}= 30,
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document.write( "or, if there is no such triple, enter the word \"none\" as your answer.
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document.write( "To simplify my writhing, I introduce new variables\r\n" );
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document.write( " a =  ; b =  ; c =  .\r\n" );
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document.write( "Then from the given conditions, I have\r\n" );
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document.write( " a + b + c = 10 (1)\r\n" );
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document.write( " a^2 + b^2 + c^2 = 38 (2)\r\n" );
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document.write( " ab + ac + bc = 30. (3)\r\n" );
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document.write( "Square equation (1) and use this identity (a+b+c)^2 = a^2 + b^2 + c^2 + 2ab + 2ac + 2bc.\r\n" );
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document.write( "You will get then\r\n" );
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document.write( " 10^2 = 38 + 2*30, or\r\n" );
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document.write( " 100 = 38 + 60 = 98. (4)\r\n" );
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document.write( "Since equality (4) is IMPOSSIBLE, it means that the given system of equations HAS NO solution.\r\n" );
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document.write( "Proved and solved.\r
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document.write( "Explained and completed.\r
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