document.write( "Question 885653: sqrt(2x+1) + 4th root(2x-1) = sqrt(x-1) + sqrt(x^2-2x+3) \n" ); document.write( "
Algebra.Com's Answer #535344 by Edwin McCravy(20060) $\"\"$ $\"About$
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document.write( "We wonder if each of the radical expressions on the left equals to one of \r\n" );
document.write( "the radical expressions on the right.\r\n" );
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document.write( "We try the first expression on left equaling to the first\r\n" );
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document.write( "We check to see if that makes the other radicals equal:\r\n" );
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document.write( "No, those are not equal.  So 2 is not a solution.\r\n" );
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document.write( "So we try the first expression on the\r\n" );
document.write( "left equaling to the second expression on the right:\r\n" );
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document.write( "We could solve this by the quadratic formula but there \r\n" );
document.write( "would be no use unless also the second expression on the left\r\n" );
document.write( "equals the first expression on the right.  So we'll wait and \r\n" );
document.write( "see if that can be the case:\r\n" );
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document.write( "To see if that can be the case, we set those equal:\r\n" );
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document.write( "Square both sides. The square of a fourth root equals\r\n" );
document.write( "the square root:\r\n" );
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document.write( "Square both sides again:\r\n" );
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document.write( "We're in luck! That just happens to be the same \r\n" );
document.write( "quadratic, so we will solve it by the quadratic formula:\r\n" );
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document.write( "We must check solutions of equations containing radicals,\r\n" );
document.write( "as they often have extraneous answers.\r\n" );
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document.write( "All the roots in the equation have even indices, so the\r\n" );
document.write( "four radicands must not be negative. \r\n" );
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document.write( "We only need to make sure that there are no negative \r\n" );
document.write( "numbers under any of the radicals:\r\n" );
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document.write( "The first radicand is\r\n" );
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document.write( "Substituting \r\n" );
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document.write( ", that's positive.  So far so good.\r\n" );
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document.write( "Substituting \r\n" );
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document.write( ", that's positive also.  OK.\r\n" );
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document.write( "The second radicand is\r\n" );
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document.write( "Substituting \r\n" );
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document.write( ", that's positive\r\n" );
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document.write( "Substituting \r\n" );
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document.write( ", that's positive also.\r\n" );
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document.write( "The third radicand is\r\n" );
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document.write( "Substituting \r\n" );
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document.write( ", that's positive\r\n" );
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document.write( "Substituting \r\n" );
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document.write( "Oh oh, that's negative, so \r\n" );
document.write( "is an extraneous answer and must be discarded.\r\n" );
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document.write( "Thus the only solution possible is , but we\r\n" );
document.write( "must make sure it gives a positive radicand in the last \r\n" );
document.write( "expression on the right:\r\n" );
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document.write( "The fourth radicand is\r\n" );
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document.write( "Substituting \r\n" );
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document.write( ", that's positive.\r\n" );
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document.write( "So there is one solution:\r\n" );
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document.write( "Edwin
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