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February 20th, 2018, 01:44 PM   #1
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System equation

Solve the system
$\displaystyle x^2 + x -1=y$
$\displaystyle y^2 +y-1=z$
$\displaystyle z^2 +z-1=x$
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February 20th, 2018, 01:53 PM   #2
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sum of the three equations ...

$x^2+y^2+z^2 + x + y + z - 3 = y+z+x$

$x^2+y^2+z^2 = 3$

sphere of radius $r=\sqrt{3}$ centered at the origin ...

https://www.wolframalpha.com/input/?...E2%2Bz%5E2%3D3
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February 20th, 2018, 02:15 PM   #3
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Quote:
Originally Posted by skeeter View Post
sum of the three equations ...

$x^2+y^2+z^2 + x + y + z - 3 = y+z+x$

$x^2+y^2+z^2 = 3$

sphere of radius $r=\sqrt{3}$ centered at the origin ...

https://www.wolframalpha.com/input/?...E2%2Bz%5E2%3D3
This is a start. However, the answer has to be specific points on the surface of the sphere. Symmetry immediately leads to (1,1,1) and(-1,-1,-1) as solutions. I don't believe there are any others.
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February 20th, 2018, 03:21 PM   #4
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If the solutions don't have to be real, there are more solutions.
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February 20th, 2018, 03:56 PM   #5
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it ends up being the roots of a degree 6 polynomial
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February 20th, 2018, 08:26 PM   #6
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If $x$ isn't real, it's a zero of $x^6 + 4x^5 + 5x^4 + 2x^3 + 1$.
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