SOLUTION: If x+(1/x) = -3, find the value of x-(1/x)

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Question 1105837: If x+(1/x) = -3, find the value of x-(1/x)
Found 2 solutions by josgarithmetic, ikleyn:
Answer by josgarithmetic(39625) About Me  (Show Source):
You can put this solution on YOUR website!
Already given. You already were given that the expression is -3.

Answer by ikleyn(52847) About Me  (Show Source):
You can put this solution on YOUR website!
.
Solution 1 

If  x + 1%2Fx%29 = -3,  then (square both sides) 

%28x%2B1%2Fx%29%5E2 = 9  ====>

x%5E2 + 2 + 1%2Fx%5E2 = 9  ====>

x%5E2 + 1%2Fx%5E2 = 9-2 = 7  ====>

x%5E2 - 2 + 1%2Fx%5E2 = 7 - 2 = 5  ====>

%28x-1%2Fx%29%5E2 = 5  ====>  x - 1%2Fx = +/- sqrt%285%29.


And both values  sqrt%285%29 and -sqrt%285%29 are achievable.


Solution 2 

If  x + 1%2Fx%29 = -3,  then 

x%5E2+%2B+3x+%2B+1 = 0  ====>

x%5B1%2C2%5D = %28-3+%2B-+sqrt%289-4%29%29%2F2 = %28-3+%2B-+sqrt%285%29%29%2F2.



1.  If  x = %28-3+%2B+sqrt%285%29%29%2F2,  then

    1%2Fx = 1%2F%28-3+%2B+sqrt%285%29%29%2F2 = 2%2F%28-3+%2B+sqrt%285%29%29 = 2%2A%28-3-sqrt%285%29%29%2F%289-5%29 = %28-3%2Bsqrt%285%29%29%2F2.


    Then  x - 1%2Fx = %28-3+%2B+sqrt%285%29%29%2F2 - %28-3+-+sqrt%285%29%29%2F2 = sqrt%285%29.



2.  If  x = %28-3+-+sqrt%285%29%29%2F2,  then similarly

    1%2Fx = 1%2F%28-3+-+sqrt%285%29%29%2F2 = 2%2F%28-3+-+sqrt%285%29%29 = 2%2A%28-3%2Bsqrt%285%29%29%2F%289-5%29 = %28-3%2Bsqrt%285%29%29%2F2.


    Then  x - 1%2Fx = %28-3+-+sqrt%285%29%29%2F2 - %28-3+%2B+sqrt%285%29%29%2F2 = -sqrt%285%29.

Solved (two times with the identical answers !).


The answer by @josgarithmetic is   I N C O R R E C T.

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For many similar solved problems see the lessons
    - HOW TO evaluate expressions involving  %28x+%2B+1%2Fx%29,  %28x%5E2%2B1%2Fx%5E2%29,  %28x%5E3+%2B+1%2Fx%5E3%29  and  %28x%5E5%2B1%2Fx%5E5%29
    - Advanced lesson on evaluating expressions
    - HOW TO evaluate functions of roots of a quadratic equation
in this site.


Also,  you have this free of charge online textbook in ALGEBRA-I in this site
    - ALGEBRA-I - YOUR ONLINE TEXTBOOK.

The referred lessons are the part of this online textbook under the topic "Evaluation, substitution".


Save the link to this online textbook together with its description

Free of charge online textbook in ALGEBRA-I
https://www.algebra.com/algebra/homework/quadratic/lessons/ALGEBRA-I-YOUR-ONLINE-TEXTBOOK.lesson

to your archive and use it when it is needed.