document.write( "Question 1085982: √(x+2√(x-1)) + √(x-2(√x-1))=x-1 \n" ); document.write( "

Algebra.Com's Answer #700215 by KMST(5328) $\"\"$ $\"About$

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$\"sqrt%28x%2B2sqrt%28x-1%29%29%2Bsqrt%28x-2sqrt%28x-1%29%29=x-1\"$
\n" ); document.write( "If those expressions have to be defined,
\n" ); document.write( "it must be $\"x-1%3E=0\"$ .
\n" ); document.write( "There are too many square roots for me to write there,
\n" ); document.write( "so I am going to say that
\n" ); document.write( " $\"y=sqrt%28x-1%29%3E=0\"$ to make the expressions simpler.
\n" ); document.write( "If $\"y=sqrt%28x-1%29\"$ , then
\n" ); document.write( " $\"y%5E2=x-1\"$ , and $\"y%5E2%2B1=x\"$ .
\n" ); document.write( "With that change of variable, the equation turns into
\n" ); document.write( " $\"sqrt%28y%5E2%2B1%2B2y%29%2Bsqrt%28y%5E2%2B1-2y%29=y%5E2\"$
\n" ); document.write( " $\"sqrt%28%28y%2B1%29%5E2%29%2Bsqrt%28%28y-1%29%5E2%29=y%5E2\"$
\n" ); document.write( "If $\"y%3E=1\"$ , then those square roots are
\n" ); document.write( " $\"y%2B1%3E=2\"$ and $\"y-1%3E=0\"$ .
\n" ); document.write( "In that case the equation simplifies to
\n" ); document.write( " $\"y%2B1%2By-1=y%5E2\"$ ,
\n" ); document.write( " $\"2y=y%5E2\"$ ,
\n" ); document.write( "and since $\"y%3E=1\"$ , we can divide both sides by y to get
\n" ); document.write( " $\"2=y\"$ .
\n" ); document.write( "That would mean $\"sqrt%28x-1%29=2\"$ ,
\n" ); document.write( " $\"x-1=4\"$ , and $\"highlight%28x=5%29\"$ .
\n" ); document.write( "
\n" ); document.write( "There is also the possibility that $\"y%3C1\"$ .
\n" ); document.write( "In that case, $\"y-1%3C0\"$ , and
\n" ); document.write( " $\"sqrt%28%28y-1%29%5E2%29=-%28y-1%29=1-y\"$
\n" ); document.write( "From the beginning we knew that $\"y%3E=0\"$ ,
\n" ); document.write( "so $\"0%3C=y%3C1\"$ ,
\n" ); document.write( " $\"y%2B1%3E0\"$ ,
\n" ); document.write( " $\"sqrt%28%28y%2B1%29%5E2%29=y%2B1\"$ , and
\n" ); document.write( "the equation simplifies to
\n" ); document.write( " $\"y%2B1%2B1-y=y%5E2\"$ ,
\n" ); document.write( " $\"2=y%5E2\"$ .
\n" ); document.write( "However, that equation has no solutions with $\"0%3C=y%3C1\"$ ,
\n" ); document.write( "so there are no more solutions.\r
\n" ); document.write( "\n" ); document.write( " \n" ); document.write( "

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