Algebra.Com's Answer #692861 by ikleyn(52781)![\"\"](\"/images/stars/stars-13.gif\")  
You can put this solution on YOUR website!
.
\n" );
document.write( ".
\n" );
document.write( "Find an ordered triple (x,y,z) of real numbers satisfying the system of equations\r
\n" );
document.write( "\n" );
document.write( "sqrt{x} + sqrt{y} + sqrt{z} = 10,
\n" );
document.write( "x + y + z = 38,
\n" );
document.write( "sqrt{xyz} = 30,\r
\n" );
document.write( "\n" );
document.write( "or, if there is no such triple, enter the word \"none\" as your answer.
\n" );
document.write( "~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\r
\n" );
document.write( "
\n" );
document.write( "\n" );
document.write( "\r\n" );
document.write( "Let me introduce new variables u =  , v =  , and w =  .\r\n" );
document.write( "\r\n" );
document.write( "Then I can rewrite the given equations in this form:\r\n" );
document.write( "\r\n" );
document.write( "u + v + w = 10, (1)\r\n" );
document.write( "\r\n" );
document.write( " = 38, (2)\r\n" );
document.write( "\r\n" );
document.write( "u*v*w = 30. (3)\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "From (1) and (2)\r\n" );
document.write( "\r\n" );
document.write( " = 100 =  =  = 38 + 2(uv + uw + vw),\r\n" );
document.write( "\r\n" );
document.write( "hence, 2(uv + uw + vw) = 100 - 38 = 62, and uv + uw + vw = 31.\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "Now I can rewrite the system (1),(2),(3) in the form\r\n" );
document.write( "\r\n" );
document.write( "u + v + w = 10, (1')\r\n" );
document.write( "\r\n" );
document.write( "uv + uw + vw = 31, (2')\r\n" );
document.write( "\r\n" );
document.write( "u*v*w = 30. (3')\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "It means ( !! The Vieta's formulas !! ) that u, v ans w are the roots of this cubical equation\r\n" );
document.write( "\r\n" );
document.write( "X^3 - 10X^2 + 31X - 30 = 0. (4)\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "Now you can apply the \"Rational roots Theorem\" which says that the rational (in particular, integer) roots of the equation (4) \r\n" );
document.write( "are among the divisors of the constant term 30.\r\n" );
document.write( "\r\n" );
document.write( "Or even better, you can make a plot of the left side polynomial which (the plot) will say you a lot about the roots:\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( " \r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "Plot y =  \r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "From the plot, it is clearly seen that the roots u, v and w are 2, 3 and 5 (from smaller to greater).\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "You can check it manually.\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "Hence, the solution to the original system is (x,y,z) = (4,9,25) and all possible permutations.\r\n" );
document.write( " \r
\n" );
document.write( "\n" );
document.write( "Isn't this solution is beautiful ?\r
\n" );
document.write( "
\n" );
document.write( "
\n" );
document.write( "\n" );
document.write( "The tutor \"Alan\" asked in his post, what we can learn from this problem ?\r
\n" );
document.write( "
\n" );
document.write( "
\n" );
document.write( "\n" );
document.write( "The answer is: THIS BEAUTIFUL SOLUTION and THIS BEAUTIFUL METHOD.\r
\n" );
document.write( "
\n" );
document.write( "
\n" );
document.write( "\n" );
document.write( "I just had a chance to solve one similar problem before. See the lesson\r
\n" );
document.write( "\n" );
document.write( " - Solving systems of non-linear algebraic equations with symmetric functions of unknowns \r
\n" );
document.write( "\n" );
document.write( "in this site.\r
\n" );
document.write( "
\n" );
document.write( "
\n" );
document.write( "\n" );
document.write( " \n" );
document.write( " |