SOLUTION: Please solve the following: if x=3+2*root2 then what is x-1/x?

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Question 938273: Please solve the following: if x=3+2*root2 then what is x-1/x?
Answer by josgarithmetic(39617) About Me  (Show Source):
You can put this solution on YOUR website!
Have tried making the substitution for x and then simplifying?

Do you mean as written, x-1/x which would appear properly as x-1%2Fx
or do you really mean,
(x-1)/x which would look properly as %28x-1%29%2Fx
?

-
Clarification given. x-1%2Fx was meant, exactly as originally given.

Best(maybe) to simplify the given expression first into one single rational expression. THEN substitute the given numerical expression for x, and simplify.
cross%28x%28x%2Fx%29-1%2Fx%29
cross%28%28x%5E2-1%29%2Fx%29
cross%28%283%2B2%2Asqrt%282%29-1%29%2F%283%2B2%2Asqrt%28x%29%29%29
(2+2sqrt(2))/(3+2sqrt(2))
I interrupt this process to do it with the originally given expression instead...


THIS IS BETTER
3%2B2sqrt%282%29-1%2F%283%2B2sqrt%282%29%29
Next, rationalize the denominator of the partial single fraction.


3%2B2sqrt%282%29-%283-2sqrt%282%29%29%2F%289-4%2A2%29, using difference of squares and combined computation
3%2B2sqrt%282%29-%283-2sqrt%282%29%29%2F1
3%2B2sqrt%282%29-%283-2sqrt%282%29%29
.
highlight%284sqrt%282%29%29