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Question Number 92219 by ajfour last updated on 05/May/20

Commented by ajfour last updated on 05/May/20

$I f t h e r e d c u r v e h a s e q u a t i o n$ $y = x^{2} - b x + c = 0 \forall b, c > 0, t h e n$ $f i n d e q u a t i o n o f u p p e r c u b i c$ $i n t h e f o r m y = {p x}^{3} + q x + r .$ $H e n c e f i n d \frac{q}{p}, \frac{r}{p} .$ $T h u s w e c a n d e t e r m i n e t h e$ $c o m m o n r o o t b y u s i n g j u s t$ $t b e C a r d a n o f o r m u l a .$ $h e n c e a l l t h e r o o t s o f t h e r e d$ $c u b i c c u r v e e v e n w h e n t h e$ $d i s c r i m i n a n t : \frac{c^{2}}{4} - \frac{b^{3}}{27} < 0 .$
Commented by ajfour last updated on 05/May/20

$(w i t h o u t t h e u s e o f t r i g o n o m e t r i c$ $s o l u t i o n o f a c u b i c e q u a t i o n)$
Commented by ajfour last updated on 05/May/20

$b u t n o, t h i s a g a i n f a i l s . .!$
Commented by MJS last updated on 05/May/20

$anyway, the black curve is not unique$
Commented by ajfour last updated on 06/May/20

$\frac{q}{p} a n d \frac{r}{p} a r e u n i q u e, S i r!$
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