find-the-real-solution-of-following-equation-system-x-2-xy-y-2-p-y-2-yz-z-2-q-z-2-zx-x-2-r-with-p-q-r-gt-0-

Question Number 180603 by mr W last updated on 14/Nov/22

f i n d t h e r e a l s o l u t i o n o f f o l l o w i n g

e q u a t i o n s y s t e m :

x^{2} + x y + y^{2} = p

y^{2} + y z + z^{2} = q

z^{2} + z x + x^{2} = r

w i t h p, q, r > 0

Commented by mr W last updated on 14/Nov/22

t h e g e n e r a l s o l u t i o n i s

λ = \sqrt{(\sqrt{p} + \sqrt{q} + \sqrt{r}) (- \sqrt{p} + \sqrt{q} + \sqrt{r}) (\sqrt{p} - \sqrt{q} + \sqrt{r}) (\sqrt{p} + \sqrt{q} - \sqrt{r})}

x = \pm \frac{\sqrt{6} λ + 3 \sqrt{2} (p - q + r)}{6 \sqrt{p + q + r + \sqrt{3} λ}}

y = \pm \frac{\sqrt{6} λ + 3 \sqrt{2} (p + q - r)}{6 \sqrt{p + q + r + \sqrt{3} λ}}

z = \pm \frac{\sqrt{6} λ + 3 \sqrt{2} (- p + q + r)}{6 \sqrt{p + q + r + \sqrt{3} λ}}

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