document.write( "Question 1066714: The number obtained by adding the squares of a number and it's reciprocal is greater by 7/4 than the sum of the number and it's reciprocal. Find the number \n" ); document.write( "

Algebra.Com's Answer #681929 by KMST(5328) $\"\"$ $\"About$

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I interpret the wording of your question as
\n" ); document.write( "find $\"x\"$ such that
\n" ); document.write( " $\"x%5E2%2B%281%2Fx%29%5E2+=+x%2B1%2Fx%2B7%2F4\"$
\n" ); document.write( "Let us say that $\"x%2B1%2Fx=y\"$ .
\n" ); document.write( "Then, $\"y%5E2=x%5E2%2B%281%2Fx%29%5E2%2B2x%281%2Fx%29\"$ ,
\n" ); document.write( " $\"y%5E2=x%5E2%2B%281%2Fx%29%5E2%2B2\"$ , and
\n" ); document.write( " $\"x%5E2%2B%281%2Fx%29%5E2=y%5E2-2\"$ .
\n" ); document.write( "With that, we can re-write the equation as
\n" ); document.write( " $\"y%5E2-2=y%2B7%2F4\"$ .
\n" ); document.write( "Multiplying both sides of the equal sign times $\"4\"$ , we get
\n" ); document.write( " $\"4y%5E2-8=4y%2B7\"$ ,
\n" ); document.write( "which simplifies to
\n" ); document.write( " $\"4y%5E2-4y-15=0\"$
\n" ); document.write( "Either applying the quadratic formula, or factoring,
\n" ); document.write( "or \"completing the square\", we can find the solutions to be
\n" ); document.write( " $\"y=5%2F2\"$ and $\"y=-3%2F2\"$ .
\n" ); document.write( " $\"x%2B1%2Fx=-3%2F2\"$ does not have real solutions,
\n" ); document.write( "but we can find real solutions for
\n" ); document.write( " $\"x%2B1%2Fx=5%2F2\"$ .
\n" ); document.write( "Multiplying both sides of the equal sign times $\"2x\"$ ,
\n" ); document.write( "we get the equivalent equation
\n" ); document.write( " $\"2x%5E2%2B2x=5\"$ <---> $\"2x%5E2%2B2x-5=0\"$ ,
\n" ); document.write( "whose solutions are
\n" ); document.write( " $\"highlight%28x=2%29\"$ and (of course) $\"highlight%28x=1%2F2%29\"$ . \n" ); document.write( "

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