SOLUTION: If x^2 + y^2 = 2 and xy = -1, then x^3 + y^3 = ?

Question 985582: If x^2 + y^2 = 2 and xy = -1, then x^3 + y^3 = ?
Answer by ikleyn(52803)   (Show Source): You can put this solution on YOUR website!

You have a system of two equations (non-linear) in two unknowns

.

Express    from the second equation = and substitute into the first equation. You will have
+ = .

Introduce = . You will have

+ = ,    - + = ,    = ,    = .

Hence, x = = +/- 1.

Correspondingly, y = -/+ 1.

Therefore, the pairs (x,y) = (1, -1) and (x,y) = (-1,1) are the solutions.   There are no other solutions.

It implies that    + = 0.

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