solve-for-a-b-c-d-R-a-2-b-c-b-2-c-d-c-2-d-a-d-2-a-b-

Question Number 54786 by behi83417@gmail.com last updated on 10/Feb/19

s o l v e f o r : a, b, c, d \in R

a^{2} = b + \sqrt{c}

b^{2} = c + \sqrt{d}

c^{2} = d + \sqrt{a}

d^{2} = a + \sqrt{b}

Commented by mr W last updated on 11/Feb/19

a = b = c = d = 0 o r {(\frac{\sqrt[3]{9 + \sqrt{69}} + \sqrt[3]{9 - \sqrt{69}}}{\sqrt[3]{18}})}^{2} \approx 1.7548 ?

Commented by behi83417@gmail.com last updated on 11/Feb/19

r i g h t a n s w e r s i r . p l e a s e p o s t y o u r

s o l u t i o n . t h a n k y o u v e r y m u c h .

Commented by mr W last updated on 11/Feb/19

d u e t o s y m m e t r y : a = b = c = d = t^{2}

t^{4} = t^{2} + t

t (t^{3} - t - 1) = 0

\Rightarrow t = 0

\Rightarrow t^{3} - t - 1 = 0

l e t t = u + v

{(u + v)}^{3} - 3 u v (u + v) - (u^{3} + v^{3}) = 0

u^{3} v^{3} = \frac{1}{27}

u^{3} + v^{3} = 1

x^{2} - x + \frac{1}{27} = 0

u^{3} (v^{3}) = \frac{1}{2} (1 \pm \sqrt{1 - \frac{4}{27}}) = \frac{1}{18} (9 \pm \sqrt{69})

u = \frac{\sqrt[3]{9 + \sqrt{69}}}{\sqrt[3]{18}}

v = \frac{\sqrt[3]{9 - \sqrt{69}}}{\sqrt[3]{18}}

\Rightarrow t = \frac{\sqrt[3]{9 + \sqrt{69}} + \sqrt[3]{9 - \sqrt{69}}}{\sqrt[3]{18}}

a = b = c = d = t^{2} = 0 o r {(\frac{\sqrt[3]{9 + \sqrt{69}} + \sqrt[3]{9 - \sqrt{69}}}{\sqrt[3]{18}})}^{2}

Commented by behi83417@gmail.com last updated on 12/Feb/19

t h a n k s a g a i n s i r . i t i s p e r f e c t .

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

s y m m e t r y ?

Commented by mr W last updated on 12/Feb/19

i a m n o t s u r e i f t h e r e a r e o t h e r s o l u t i o n s .

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