SOLUTION: If x^2 + 1/x^2 =102, find the value of x^3 - 1/x^3

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Question 1086799: If x^2 + 1/x^2 =102, find the value of x^3 - 1/x^3
Found 2 solutions by rothauserc, ikleyn:
Answer by rothauserc(4718)   (Show Source): You can put this solution on YOUR website!
x^2 + 1/x^2 = 102
:
x^4 + 1 = 102x^2, for x not = 0
:
x^4 -102X^2 +1 = 0
:
let u = x^2, then
:
u^2 -102u = -1
:
complete the square
:
u^2 -102u + 2601 = 2601 -1
:
(u -51)^2 = 2600
:
take square root of both sides of =
:
u - 51 = 50.99
:
u = 51 + 50.99 = 101.99
:
u = x^2, then
:
x^2 = 101.99
:
x = + or - 10.099
:
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x^3 - 1/x^3 = ?
:
1) (10.099)^3 - 1/(10.099)^3 = 1029.994
:
2) (-10.099)^3 - 1/(-10.099)^3 = −1029.994
:
check answers for x
:
x^2 + 1/x^2 = 102
:
(10.099)^2 + 1/(10.099)^2 = 102
101.999 = 102
:
(-10.099)^2 + 1/(-10.099)^2 = 102
101.999 = 102
:
our answer checks
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:

Answer by ikleyn(52787)   (Show Source): You can put this solution on YOUR website!
.
If  = 102,  then   - 2 +  = 102 - 2 = 100, 

or   = 100,  which implies    = 10  or   = -10.


If   = 10  then

     = 1000 =  =  -  =  -  =  - 3*10.


It implies   = 1030.



If   = -10  then

     = -1000 =  =  -  =  -  =  - 3*(-10) =  + 30.


It implies   = -1000 - 30 = -1030.

Answer.  Under the given condition,  the expression    may have two values:  1030  or  -1030.


Solved.


Two lessons to learn from this solution

    1.  You do not need to solve equations to get the answer.

    2.  The approach the tutor @rothauserc took to solve the problem is WRONG.



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