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\n" ); document.write( "\n" ); document.write( "(A) x^2 - 3x - sqrt{x^2 - 3x} = 2\r
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\n" ); document.write( "\n" ); document.write( "(B) 3x^(4/3) + 5x^(2/3) = 0
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\n" ); document.write( "\n" ); document.write( " I will solve equation (A).\r
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document.write( " (A) x^2 - 3x - sqrt{x^2 - 3x} = 2.\r\n" );
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document.write( "The domain of equation (A) is the set of real numbers x such that\r\n" );
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document.write( " x^2 - 3x >= 0, x*(x-3) >= 0, x <= 0 OR x >= 3. (1)\r\n" );
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document.write( "Introduce new variable t = 
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document.write( "Then equation (A) takes the form\r\n" );
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document.write( " t^2 - t = 2. (2)\r\n" );
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document.write( "We consider 
as a non-negative value; \r\n" );
document.write( "so, we look for the solution of equation (2) with t >= 0.\r\n" );
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document.write( "Write equation (2) in the standard form quadratic equation\r\n" );
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document.write( " t^2 - t - 2 = 0.\r\n" );
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document.write( "Factor left side\r\n" );
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document.write( " (t-2)*(t+1) = 0.\r\n" );
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document.write( "Two roots are t= 2 and t= -1.\r\n" );
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document.write( "We are looking for non-negative t, so we ignore t= -1.\r\n" );
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document.write( "Now consider t= 2\r\n" );
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document.write( "It means = 2, x^2 - 3x = 2^2 , x^2 - 3x = 4, \r\n" );
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document.write( " x^2 - 3x - 4 = 0, (x-4)*(x+1) = 0, x= 4 or x= -1.\r\n" );
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document.write( " These potential solutions are in the domain of equation (A), \r\n" );
document.write( " so they are valid solutions to equation (A).\r\n" );
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document.write( "ANSWER. Equation (A) has two solutions: x= -1 and x= 4.\r\n" );
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document.write( "You may check that the answer is correct, by substituting these values into equation (A). \r\n" );
document.write( "\r\n" ); document.write( "\n" ); document.write( "Equation (A) is solved.\r
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