Algebra.Com's Answer #833058 by ikleyn(52786)![\"\"](\"/images/stars/stars-13.gif\")  
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document.write( "Solve for x this equation x^2 + 3x + sqrt{x^2 + 3x} = 6.
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document.write( " The method to solve is introducing new variable.\r
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document.write( "(1) Notice that the expression under the square root must be non-negative\r\n" );
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document.write( " x^2 + 3x >= 0, or, equivalently, x*(x+3) >=0,\r\n" );
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document.write( " which means that either x >= 0 or x <= -3.\r\n" );
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document.write( " Thus the domain is (-oo,-3] U [0,oo).\r\n" );
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document.write( " Also, notice that we consider positive values of  , only, so y >= 0.\r\n" );
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document.write( "(2) Introduce new variable y =  . \r\n" );
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document.write( " Then y^2 = x^2 + 3x, and the given equation takes the form\r\n" );
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document.write( " y^2 + y = 6, or, equivalently,\r\n" );
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document.write( " y^2 + y - 6 = 0\r\n" );
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document.write( " Factor left sides\r\n" );
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document.write( " (y+3)*(y-2) = 0\r\n" );
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document.write( " The roots are y= -3 ans y= 2.\r\n" );
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document.write( " Since we consider only positive values of y in this problem, \r\n" );
document.write( " we analyse below the case y= 2 ONLY.\r\n" );
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document.write( "(3) If y= 2, then\r\n" );
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document.write( " = 2.\r\n" );
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document.write( " Square both sides, simplify and find x\r\n" );
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document.write( " x^2 + 3x = 4\r\n" );
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document.write( " x^2 + 3x - 4 = 0\r\n" );
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document.write( " (x+4)*(x-1) = 0\r\n" );
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document.write( " Two roots are x= -4, x= 1. \r\n" );
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document.write( "Solved, and the obtained answer coincides with that in your textbook.\r
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document.write( "Memorize the method of the solution: it is introducing new variable
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document.write( "and accurate analysis of the domains of the variables x and y.\r
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