Solving a Tricky Nonlinear Equation System: A Quest for Closed Form Solutions

[n7imo]

I'm trying to find a closed form (an algebraic solution) for the following system:

x² - y² = 5
x + y = xy

It's a bit tricky but I manage to end up with the quartic equation:
x^4 - 2x^3 + 5x^2 -10x + 5 =0
And this is where I get stuck looking for a closed form root.
Any suggestion would be appreciated

 

[Eucliddo]

trial and error of synthetic division.

 

[SteamKing]

I'm trying to find a closed form (an algebraic solution) for the following system:

x² - y² = 5
x + y = xy

It's a bit tricky but I manage to end up with the quartic equation:
x^4 - 2x^3 + 5x^2 -10x + 5 =0
And this is where I get stuck looking for a closed form root.
Any suggestion would be appreciated

You won't find any integer or rational solutions. The general solution to a fourth degree equation is pretty daunting.

https://en.wikipedia.org/wiki/Quartic_function

This particular equation has two real and a pair of complex-conjugate roots.

BTW, I checked your algebra in reducing your system of equations to one equation in x. I think you have some mistakes there, since I don't obtain your particular quartic equation.

In any event, the resulting quartic still has two real and a pair of complex-conjugate solutions, none of which are nice integers or rationals.

I used Wolfram Alpha to solve for the roots. It's much easier than anything else.

 

[n7imo]

You won't find any integer or rational solutions. The general solution to a fourth degree equation is pretty daunting.

https://en.wikipedia.org/wiki/Quartic_function

This particular equation has two real and a pair of complex-conjugate roots.

BTW, I checked your algebra in reducing your system of equations to one equation in x. I think you have some mistakes there, since I don't obtain your particular quartic equation.

In any event, the resulting quartic still has two real and a pair of complex-conjugate solutions, none of which are nice integers or rationals.

I used Wolfram Alpha to solve for the roots. It's much easier than anything else.

Indeed, the right resulting quartic equation is x^4 - 2x^3 - 5x^2 -10x - 5 =0. I used Cardano and Lagrange method to find the real roots, but their form is very ugly.
Actually I got this equation while trying to solve a simple geometrical problem. I'll post it today on a new thread, I'm interested in finding a simpler method to solving it since mine leads to a quartic equation.

Thanks for the contribution.