Solve simultaneous eqs. `x^3=5x+y` ; `y^3=x+5y` ?

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embizze | High School Teacher | (Level 2) Educator Emeritus

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Solve the system `x^3=5x+y,y^3=x+5y` :

Solving the first equation for `y` we get `y=x^3-5x`. Substituting into the second equation we get `(x^3-5x)^3=x+5(x^3-5x)` .

Expanding and collecting like terms we get:




Thus x=0,y=0 is one solution. We solve the 8th degree polynomial:

Using the rational root theorem we find -2 and 2 are roots. Using synthetic division or polynomial long division we have:



There are no rational roots of `x^6-11x^4+31x^2-6` , so we try `+-sqrt(6)` which work. Using synthetic division we get :


We treat `x^4-5x^2+1` as a quadratic in `x^2` ; using completing the square we get:

`x^4-5x^2+1=0 ==> (x^2-5/2)^2-25/4+1=0`





Thus the x values for the solution are `-2,0,2,-sqrt(6),sqrt(6),+-sqrt(1/2(5+-sqrt(21)))`


The solutions to the system are: