SOLUTION: 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

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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
Answer by KMST(5328) About Me  (Show Source):
You can put this solution on YOUR website!
I interpret the wording of your question as
find x such that
x%5E2%2B%281%2Fx%29%5E2+=+x%2B1%2Fx%2B7%2F4
Let us say that x%2B1%2Fx=y .
Then, y%5E2=x%5E2%2B%281%2Fx%29%5E2%2B2x%281%2Fx%29 ,
y%5E2=x%5E2%2B%281%2Fx%29%5E2%2B2 , and
x%5E2%2B%281%2Fx%29%5E2=y%5E2-2 .
With that, we can re-write the equation as
y%5E2-2=y%2B7%2F4 .
Multiplying both sides of the equal sign times 4 , we get
4y%5E2-8=4y%2B7 ,
which simplifies to
4y%5E2-4y-15=0
Either applying the quadratic formula, or factoring,
or "completing the square", we can find the solutions to be
y=5%2F2 and y=-3%2F2 .
x%2B1%2Fx=-3%2F2 does not have real solutions,
but we can find real solutions for
x%2B1%2Fx=5%2F2 .
Multiplying both sides of the equal sign times 2x ,
we get the equivalent equation
2x%5E2%2B2x=5<--->2x%5E2%2B2x-5=0 ,
whose solutions are
highlight%28x=2%29 and (of course) highlight%28x=1%2F2%29 .